diff --git "a/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt" "b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt" @@ -0,0 +1,848 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf,len=847 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='00753v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='IT] 2 Jan 2023 POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES REZA DASTBASTEH AND KHALIL SHIVJI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We give a polynomial representation for additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This repre- sentation will be applied to uniquely present each additive cyclic code by at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We determine the generator polynomials of all different additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Keywords: additive cyclic codes, quantum code, self-orthogonal codes, self-dual codes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Introduction Quantum error-correcting codes, or simply quantum codes, are used in quantum computation to protect quantum information from corruption by noise (decoherence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A general framework of quantum codes is provided in [9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Throughout this paper, Fp2 is the finite field of p2 elements, where p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameters of a quantum code over Fp that encodes k logical qubits to n physical qubits and has minimum distance d is denoted by [[n, k, d]]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An important family of quantum codes with many similar properties as classical block codes is the family of quantum stabilizer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, quantum stabilizer codes are constructed using additive codes which are self-orthogonal with respect to a certain symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Several constructions of quantum stabilizer codes from various classical codes are given in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An interesting modification of the original definition of quantum stabilizer codes is by relaxing its self-orthogonality constraint [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This method enables us to construct good quantum codes using not necessarily self-orthogonal additive codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Previously, this modification was applied for the construction of new quantum codes from different families of linear codes [6, 10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes are of interest due to their rich algebraic properties and application in the construction of quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' There have been several works in the literature toward the classification of additive cyclic codes for different applications [1, 4, 7, 16, 17, 21], and also due to their connection to other families of block codes such as quasi-cyclic codes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In [16], a canonical decomposition of additive cyclic code over F4 was introduced using certain finite field extensions of F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This decomposition was applied to determine self-orthogonal and self-dual additive cyclic codes over F4 with respect to the trace inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In [3], it was shown that each additive cyclic code over F4 of length n can be generated by F2-span of at most two polynomials in F4[x]/⟨xn − 1⟩ and their cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, a criterion for the self-orthogonality of such codes with respect to the trace inner product was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Another interesting construction for a subclass of additive cyclic code, namely twisted codes, was provided in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This construction is analogous to the way linear cyclic codes are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In spite of many useful properties of twisted codes, all additive cyclic codes cannot be described using the theory of additive twisted codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 1 POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 2 In this work, we first give a canonical representation of all Fp-additive cyclic codes over Fp2 using at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our representation is more computationally friendly than the canonical representation of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation allows us to give a minimum distance lower bound for additive cyclic codes over Fp2 using the minimum distance of linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we provide a unique set of generator polynomials for each additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation of generator polynomials will be used to characterize all self-orthogonal and self-dual additive cyclic codes with respect to the symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We also determine the generator polynomials of the symplectic dual of a given additive cyclic code over Fp2, and compute nearly the self-orthogonality of each additive cyclic code using only its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This allows us to apply the nearly self-orthogonal construction of quantum codes developed in [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, we provide a list of eleven record-breaking binary quantum codes after applying the mentioned quantum construction to nearly self-orthogonal additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Furthermore, applying secondary constructions to our new quantum codes produce many more record-breaking binary codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that such new quantum codes cannot be constructed using self-orthogonal additive cyclic codes of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Section 2 briefly recalls the essential terminologies used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Section 3 gives a canonical representation of additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In fact, we follow a module theory approach to decompose each additive cyclic code using its polynomial representation in Fp2[x]/⟨xn −1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In Section 4, we compute the symplectic dual of each additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We provide the necessary and sufficient conditions for an additive cyclic code to be self-orthogonal, self-dual, or nearly self-orthogonality with respect to the symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, in Section 5, we present the parameters of our record-breaking quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Preliminaries Let ω be a primitive element of Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then the set {1, ω} forms a basis for Fp2 over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let a + bω and a′ + b′ω ∈ Fn p2, where a, a′, b, b′ ∈ Fn p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The symplectic inner product of a + bω and a′ + b′ω is defined by ⟨a + bω, a′ + b′ω⟩s = a′ · b − a · b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) An Fp-linear subspace C ⊆ Fn p2 is called a length n additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the Fp-dimension of an additive code C over Fp2 with dimFp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C ⊆ Fn p2 be an additive code over Fp2 such that dimFp(C) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we call C an (n, pk) code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The set C⊥s = {x ∈ Fn p2 : ⟨x, y⟩s = 0 for all y ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' is called the symplectic dual of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily see that C⊥s is an (n, p2n−k) additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is called self-orthogonal (respectively self-dual) if C ⊆ C⊥s (respectively if C = C⊥s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each x ∈ Fn p2, we denote the number of non-zero coordinates of x by wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the minimum weight among non-zero vectors of an additive code C is denoted by d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The connection between quantum stabilizer codes and classical additive codes was initially formulated by the independent works of Calderbank, Rains, Shor, and Sloane [3] and Gottesman [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A non-binary version of this connection is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [18, Corollary 16] Let C be an (n, pn−k) additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there exists an [[n, k, d]]p quantum stabilizer code if C is symplectic self-orthogonal, where d = min{wt(x) : x ∈ C⊥ s \\ C} if k > 0 and d = min{wt(x) : x ∈ C} if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The quantum code of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 is called pure if d = d(C⊥s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' There are several secondary constructions of quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A short list of such constructions is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [18, Section XV] Let C be an [[n, k, d]]p quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) If k > 0, then an [[n + 1, k, d]]p quantum code exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) If C is pure and n, d ≥ 2, then an [[n − 1, k + 1, d − 1]]p pure quantum code exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) If k > 1, then there exists an [[n, k − 1, d]]p quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes over Fp2 Throughout this section, we assume that n is a positive integer such that (n, p) = 1 and Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree two irreducible polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this section, we provide a canonical representation of additive cyclic codes over the field Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, we give a unique representation of each additive cyclic code over Fp2 using at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we determine the generator polynomials of all different additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, each additive cyclic code over F2 p is a linear combination of cyclic shifts of its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Such representation is also suitable for practical computations of additive cyclic codes, especially using Magma computer algebra system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' More particularly, there exists a built-in function in Magma which forms additive cyclic codes generated by two given generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' At the end of this section, we give a minimum distance lower bound for the minimum distance of additive cyclic codes over Fp2 using the minimum distance of linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An Fp-subspace C ⊆ Fn p2 is called an additive cyclic code of length n over Fp2, if for every (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−1) ∈ C, the vector (an−1, a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−2) is also a codeword of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We will use the following concepts of module theory frequently in this section, and for more details one, for example, can see [8, Chapter 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let R be a principal ideal domain and M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The annihilator of M is an ideal of R defined by {r ∈ R : rm = 0 for any m ∈ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An element m ∈ M is called a torsion element, if there exists 0 ̸= r ∈ R such that rm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The module M is called a torsion module if all of its elements are torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The following theorem, known as the primary decomposition theorem of modules, plays an important role in our representation of additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [8, Chapter 12, Theorem 7] Let R be a principal ideal domain and M be a torsion R-module with the annihilator ⟨a⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let a = u n � i=1 pai i , where u is a unit and pi is a prime element for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can decompose M as a direct sum of its submodules in the form M = n � i=1 Ni, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) where Ni = {x ∈ M : xpai i = 0} for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Each element (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−1) ∈ Fn p2 can be represented uniquely as a polynomial in Fp2[x]/⟨xn− 1⟩ in the form n−1 � i=0 aixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily verify that, under this correspondence, a length n additive cyclic codes over Fp2 is an Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f and g ∈ Fp2[x]/⟨xn −1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We fix the following notations for the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) The ideal generated by f in Fp2[x]/⟨xn − 1⟩ is denoted by ⟨f⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Equivalently it is the Fp2[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 4 (2) The Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial g is denoted by ⟨g⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A straightforward computation shows that the annihilator of Fp2[x]/⟨xn − 1⟩ as an Fp[x]- module is the ideal ⟨xn−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we can decompose xn−1 over Fp[x] as xn−1 = s � i=1 fi(x), where each fi(x) is an irreducible polynomial corresponding to a p-cyclotomic coset modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 to Fp2[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It is straightforward to see that Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x] for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We call a non-zero length n additive cyclic code C over Fp2 irreducible if for any additive cyclic code D ⊆ C, then D = {0} or D = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The next lemma shows that each Ni can be decomposed as a direct sum of two irreducible additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We determine the generator polynomial of all irreducible additive cyclic codes inside Ni and provide other useful information about additive cyclic codes inside each Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) be an irreducible divisor of xn − 1 over Fp[x] with deg(f) = k and N = ⟨(xn − 1)/f(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Let 0 ̸= r(x) ∈ N, then the set L = {r(x), xr(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , xk−1r(x)} forms a basis for ⟨r(x)⟩Fp[x] as an Fp vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let 0 ̸= C ⊊ N be an additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C has Fp-dimension k and C = ⟨r(x)⟩Fp[x] for any 0 ̸= r(x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) The additive cyclic code N can be decomposed as N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, dimFp(N) = 2k and N is linear over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) The number of irreducible additive cyclic codes inside N is 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the following set gives all the different generator polynomials of such additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A = { � (xn − 1)/f(x) �� ω + g(x) � : g(x) ∈ Fp[x], deg(g(x)) < k} ∪ {(xn − 1)/f(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Obviously L ⊆ ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose, on the contrary, that L is linearly dependent over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence we can find a polynomial 0 ̸= s(x) ∈ Fp[x] of degree less than k such that r(x)s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since (xn − 1)/f(x) | r(x) and f(x) is irreducible, we conclude that f(x) | s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' However, it is a contradiction with the fact that deg(s(x)) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This shows that L is linearly independent over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the set L ∪ {xkr(x)} is linearly dependent over Fp as this new set generates f(x)r(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In a similar fashion, one can show that {xir(x)} for k < i < n − 1 can be written as a linear combination of elements of L over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, L forms a basis for ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let 0 ̸= r(x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose in contrary that ⟨r(x)⟩Fp[x] ⊊ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there exists a polynomial s(x) ∈ C such that s(x) ̸∈ ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that ⟨r(x)⟩Fp[x] ∩ ⟨s(x)⟩Fp[x] = {0} as otherwise, by part (1), for any polynomial a(x) in the intersection, we have ⟨r(x)⟩Fp[x] = ⟨a(x)⟩Fp[x] = ⟨s(x)⟩Fp[x], which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus C = ⟨r(x)⟩Fp[x] and has dimension k over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) It is easy to see that ⟨(xn − 1)/f(x)⟩Fp[x] ∩ ⟨ω((xn − 1)/f(x)⟩Fp[x] = {0} and N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 5 Hence N has dimension 2k over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The linearity part follows immediately from the structure of its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) In order to find an additive cyclic code with Fp-dimension k, we need to choose a nonzero polynomial r(x) ∈ N to be its generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Also, any non-zero elements of ⟨r(x)⟩Fp[x] generates the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the number of additive cyclic codes with one non-zero generator inside N is 22k−1 2k−1 = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C1 and C2 be two k-dimensional additive cyclic codes inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If C1 ∩ C2 ̸= {0}, then C1 = C2 by part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Equivalently, if C1 + C2 = N, then C1 ∩ C2 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now we show that different elements of the set A generate different codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let g(x) ∈ Fp[x] such that deg(g(x)) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Clearly the additive cyclic code C1 = ⟨(xn − 1)/f(x), ((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] contains (xn − 1)/f(x) and ω(xn − 1)/f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore C1 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So ⟨(xn − 1)/f(x)⟩Fp[x] and ⟨((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] are different additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let g1(x) and g2(x) ∈ Fp[x] be two different polynomials of degree less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C = ⟨((xn − 1)/f(x))(ω + g1(x)), ((xn − 1)/f(x))(ω + g2(x))⟩Fp[x] contains (xn − 1)/f(x) and ω(xn − 1)/f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It is mainly because ⟨ � (xn − 1)/f(x) � (g1(x) − g2(x))⟩Fp[x] = ⟨(xn − 1)/f(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus C = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that the additive cyclic codes ⟨((xn − 1)/f(x))(ω + g1(x))⟩Fp[x] and ⟨((xn−1)/f(x))(ω+g2(x))⟩Fp[x] are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves that the set A contains all the different generators of irreducible additive cyclic codes inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ As we mentioned in part (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, each additive cyclic code inside ⟨(xn−1)/f(x)⟩Fp2[x] can have many different generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Through the next remark, we fix a canonical representation for each additive cyclic code inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each additive code 0 ̸= C ⊊ ⟨(xn −1)/f(x)⟩Fp2[x], we fix its generator polyno- mial inside the set A, introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3), to be “the” generator polynomial of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Similarly, the additive cyclic code C′ = ⟨(xn−1)/f(x)⟩Fp2[x] can be generated by the polynomials (xn−1)/f(x) and ω((xn − 1)/f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We call them “the” generator polynomials of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation helps to uniquely identify each additive cyclic code inside N and avoid considering the same code more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we use the result of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 and characterize all the additive cyclic codes of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that xn − 1 = s � i=1 fi(x), where fi(x) is an irreducible polynomial over Fp[x] for each 1 ≤ i ≤ s and Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then (i) we can decompose the code C as C = s � i=1 Ci, where each Ci is an additive cyclic code inside Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) we have C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x], where (a) g(x) + ωk(x) = s � i=1 gi(x) + ωki(x), (b) h(x) = s � i=1 hi(x), (c) and Ci has the generator polynomial(s) gi(x) + ωki(x) and ωhi(x) selected as dis- cussed in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 6 (iii) dimFp(C) = s � i=1 (deg(fi) × # of non-zero generators of Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) As we mentioned in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2), the following decomposition holds Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we can express C as C = �s i=1 Ci, where each Ci is an additive cyclic codes inside Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) We show that the additive cyclic codes C = s � i=1 Ci and ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First note that g(x) + ωk(x), ωh(x) ∈ C and thus ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let 1 ≤ i ≤ s be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since � (xn − 1)/fi(x) � | gi(x), ki(x), hi(x) and � (xn − 1)/fi(x) � gj(x) ≡ � (xn − 1)/fi(x) � kj(x) ≡ � (xn − 1)/fi(x) � hj(x) ≡ 0 (mod xn − 1) for any j ̸= i, we have � (xn − 1)/fi(x) �� g(x) + ωk(x) � ≡ � (xn − 1)/fi(x) �� gi(x) + ωki(x) � (mod xn − 1) and � (xn − 1)/fi(x) � ωh(x) ≡ � (xn − 1)/fi(x) �� ωhi(x) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we have Ci = ⟨gi(x)+ωki(x), ωhi(x)⟩Fp[x] = ⟨ � (xn −1)/fi(x) �� g(x)+ωk(x) � , � (xn −1)/fi(x) � ωh(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This show that s � i=1 Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (iii) Note that dimFp(C) = s � i=1 dimFp(Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, dimFp(Ci) = 0, ki, or 2ki if Ci = 0, Ci is generated by one generator polynomial, or Ci has two generator polynomials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Combining these facts with the result of part (i) completes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Through the next corollary, we characterize all the length n irreducible additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be an additive cyclic code of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is irreducible if and only if C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, there are s � i=1 (2deg(fi) + 1) many different irreducible additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The result of part (1) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 shows that C is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Conversely, let C be an irreducible additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then by part (i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 we have C = s � i=1 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since C is irreducible, we have C = Cj for some 1 ≤ j ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, since Nj is not irreducible by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (3), we conclude that C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 7 Inside each Ni, there are 2deg(fi)+1 many different one generator additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the total number of irreducible codes is s � i=1 (2deg(fi) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Henceforth, we always represent each additive cyclic code with its generator polynomials g(x) + ωk(x) and ωh(x) introduced in part (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the way we generate these polynomials is unique, and therefore each additive cyclic code has a unique set of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' From now on, we call Fp2-linear cyclic codes simply linear cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 and part (3) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 imply that C is linear if and only if g(x) = h(x) and k(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence linear cyclic codes can be easily distinguished from non-linear cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we provide a minimum distance bound for additive cyclic codes using linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, the minimum distance computation for linear codes is faster than the additive codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the following result can speed up the minimum distance computation for additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the minimum distance of a code C with d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let G(x) = xn−1 gcd(xn−1,g(x)), and let S(x) be the generator polynomial of the intersection of the length n linear cyclic code generated by k(x) and the linear cyclic code generated by h(x) over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose that D1, D2, D3, and D3 are the length n linear cyclic codes over Fp generated by g(x), gcd(k(x), h(x)), gcd(G(x)k(x), h(x)), and g(x)S(x) gcd(xn−1,k(x)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(D3), d(D4), max{d(D1), d(D2)}} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Only the following three types of codewords may appear in the code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, T2 = {ωb(x) ∈ C : 0 ̸= b(x) ∈ Fp[x]}, T3 = {a(x) + ωb(x) ∈ C : 0 ̸= a(x), 0 ̸= b(x) ∈ Fp[x]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We bound the minimum distance of C by considering the minimum distance in each of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) ∈ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can write it as f(x) = a1(x)(g(x) + ωk(x)) + b1(x)ωh(x) for some a1(x), b1(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence f(x) = a1(x)g(x) and a1(x)k(x) + b1(x)h(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that a1(x)k(x) is an element of the length n linear cyclic code over Fp generated by S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence S(x) gcd(xn−1,k(x)) | a1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In other words, f(x) = a(x)g(x) ∈ D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, let ωf1(x) ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then ωf1(x) = a1(x)(g(x)+ωk(x))+b1(x)ωh(x) for some a1(x), b1(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then a1(x)g(x) ≡ 0 (mod xn − 1) or equivalently G(x) | a1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that f1(x) = a1(x)k(x) + b1(x)h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, f1(x) ∈ D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, let a(x) + ωb(x) ∈ T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then a(x) + ωb(x) = l(x)(g(x) + ωk(x)) + m(x)ωh(x) for some l(x), m(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence a(x) ∈ D1 and b(x) ∈ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that wt(a(x) + ωb(x)) ≥ max{d(D1), d(D2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that if Di = 0 for any value 1 ≤ i ≤ 4, then we simply discard this code in the minimum distance lower bound of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For instance if D1 = 0, then the minimum distance lower bound of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4) becomes min{d(D3), d(D4), d(D2)} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The following corollary gives a modification of this result to additive cyclic codes, which are generated by only one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this result, the cyclic codes Ci are obtained from Di after POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 8 substituting h(x) with 0 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 for 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' However, the code C4 is obtained differently by considering a more direct observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C1, C2, C3, and C4 be the length n linear cyclic codes over Fp generated by polynomials g(x), k(x), xn−1 gcd(xn−1,g(x))k(x), and xn−1 gcd(xn−1,k(x))g(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(C3), d(C4), max{d(C1), d(C2)}} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' As we mentioned above, the code Ci all are obtained after applying the condition h(x) = 0 in the structure of the codes Di for 1 ≤ i ≤ 3 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since the code D4 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 is applied to bound the minimum weight of the set T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, we compute the minimum weight of T1 directly in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) ∈ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can write it as f(x) = a(x)(g(x)+ωk(x)) for some a(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence f(x) = a(x)g(x) and a(x)k(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that xn−1 gcd(xn−1,k(x)) | a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence xn−1 gcd(xn−1,k(x))g(x) | a(x) and we have f(x) ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we consider the restriction of the mentioned minimum distance bound to linear cyclic codes with the generator polynomials g(x) + ωk(x) and h(x), where k(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let E1 and E2 be the length n linear cyclic codes over Fp generated by polynomials g(x) and h(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(E1), d(E2)} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Applying the condition k(x) = 0 to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 implies that D1 = D4 = E1 and D2 = D3 = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now the result follows from the minimum distance bound of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Symplectic inner product and dual of additive cyclic codes In this section, we determine generator polynomials of the symplectic dual of a given additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we give the generator polynomials of all self-orthogonal and self- dual codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We also measure how close is a given additive cyclic code from being symplectic self- orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that p is a prime number and n is a positive integer coprime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, elements of Fp2 are represented by Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree 2 irreducible polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) we defined the symplectic inner product of two elements in Fn p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We define the symplectic inner product of two polynomials analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, for c(x) = n−1 � i=0 (ai + ωbi)xi and c′(x) = n−1 � i=0 (a′ i + ωb′ i)xi ∈ Fp2[x]/⟨xn − 1⟩, we define c(x) ∗ c′(x) = n−1 � i=0 (aib′ i − a′ ibi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Here we use a different notation for the symplectic inner product to differentiate between the vectors and polynomials as different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let c(x) = g1(x) + ωg2(x) and c′(x) = g′ 1(x) + ωg′ 2(x) be two polynomials of Fp2[x]/⟨xn − 1⟩, where g1(x), g2(x), g′ 1(x), g′ 2(x) ∈ Fp[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then c(x) ∗ c′(x) is the constant term of g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A similar argument shows that c(x) ∗ xic′(x) is the coefficient of xi in g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus if POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 9 g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) ≡ 0 (mod xn − 1), then the code generated by c′(x) lies in the symplectic dual of the code generated by c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We use this property very frequently through this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily verify that the symplectic dual of an additive cyclic code C over Fp2 is also an additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 part (ii), each additive cyclic code of length n over Fp2 can be represented uniquely as C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x], where g1(x), g2(x), h(x) ∈ Fp[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our next theorem gives a criterion for the self-orthogonality of additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The proof is very similar to that of [3, Theorem 14 part c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is self-orthogonal if and only if the following conditions are satisfied: (1) g2(x)h(x−1) ≡ 0 (mod xn − 1), (2) g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ⇒: Suppose that C is self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each 0 ≤ i ≤ n − 1, the inner product of g1(x) + ωg2(x) and xih(x) is the coefficient of xi in −g2(x)h(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since C is self- orthogonal, we have g2(x)h(x−1) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, � xi(g1(x) + ωg2(x)) � ∗ � g1(x) + ωg2(x) � is the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence, for each 0 ≤ i ≤ n − 1, the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ⇐: Conversely, the fact that g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1) implies that all the vectors inside ⟨g1(x) + ωg2(x)⟩Fp[x] are self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, since g2(x)h(x−1) ≡ 0 (mod xn − 1), we conclude that h(x) is orthogonal to all the cyclic shifts of g1(x) + ωg2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally h(x) ∗ xih(x) = 0 for each 0 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] is a symplectic self-orthogonal code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Recall that xn − 1 = s � i=1 fi(x), where each fi(x) is an irreducible polynomial in Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, as we mentioned earlier in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2), we have Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni, where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let α be a primitive n-th root of unity in a finite filed extension of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the p-cyclotomic cosets modulo n by Zi for each 1 ≤ i ≤ s in the way that fi(x) = � a∈Zi (x − αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This gives a one-to-one correspondence between the sets Ni and all the p-cyclotomic cosets modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our first goal in this section is to find the symplectic dual of a given additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In order to achieve this goal, we need a few preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the next lemma, we find the symplectic dual of each Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let 1 ≤ i ≤ s and C = Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = s � k=1 k̸=j Nk, where Zj = −Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First note that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (3) we have C = ⟨(xn−1)/fi(x), ω((xn−1)/fi(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If Zk ̸= −Zi, then fi(x) | (xn − 1)/fk(x−1) and fk(x) | (xn − 1)/fi(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we have � (xn − 1)/fi(x) �� (xn − 1)/fk(x−1) � ≡ 0 (mod xn − 1) and � (xn − 1)/fk(x) �� (xn − 1)/fi(x−1) � ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 10 Hence the symplectic inner product of each element of Ni and each element of Nk is zero by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves that s � k=1 k̸=j Nk ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that both of Ni and Nj have Fp-dimension 2 deg(fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now, the facts that dimFp(C) + dimFp(C⊥s) = 2n and dimFp(C) = 2 deg(fi) implies the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we find the symplectic dual of each irreducible additive cyclic code inside Ni for 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C ⊊ Ni be a non-zero additive cyclic code for some 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = ( s � k=1 k̸=j Nk) � ⟨g1(x) + ωg2(x)⟩Fp[x], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) where Zj = −Zi and g1(x) + ωg2(x) = � ((xn − 1)/fj(x))(s(x−1) + ω) if C = ⟨ � (xn − 1)/fi(x) �� ω + s(x) � ⟩Fp[x] (xn − 1)/fj(x) if C = ⟨(xn − 1)/fi(x)⟩Fp[x] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, one can see that s � k=1 k̸=j Nk ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that dimFp(⟨g1(x)+ωg2(x)⟩Fp[x]) = dimFp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So it is sufficient to show that C is orthogonal to g1(x) + ωg2(x) and all its cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We prove the latter statement in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First suppose that C = ⟨ � (xn−1)/fi(x) �� ω+ s(x) � ⟩Fp[x] for some s(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' To show that the codes C and ⟨g1(x) + ωg2(x)⟩Fp[x] are orthogonal, we apply Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, � (xn − 1)/fi(x) � s(x)g2(x−1) − � (xn − 1)/fi(x) � g1(x−1) ≡ � (xn − 1)/fi(x) � s(x) � (xn − 1)/fi(x) � − � (xn − 1)/fi(x) �� (xn − 1)/fi(x) � s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, suppose that C = ⟨(xn − 1)/fi(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then � (xn − 1)/fi(x) � g2(x−1) − � (xn − 1)/fi(x) � g1(x−1) ≡ � (xn − 1)/fi(x) � 0 − 0 � (xn − 1)/fj(x) � s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This shows that the code C is orthogonal to the additive cyclic code generated by g1(x)+ωg2(x) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that as we showed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, when C = ⟨ � (xn − 1)/fi(x) �� ω + s(x) � ⟩Fp[x], its symplectic inner product contains the code C′ = ⟨(xn − 1)/fj(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C′ is not in one of the forms given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In order to express the code C′ using the standard notation introduced in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4), we choose its generator to be g(x) = (xn − 1)/fj(x))(t(x) + ω), where t(x) ≡ s(x−1) (mod fj(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now it is easy to see that g(x) belongs to the set A introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4) and C′ = ⟨(xn −1)/fj(x))(t(x)+ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we combine the results of the previous two lemmas and the result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 to determine generator polynomials of the symplectic dual for any additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = ⟨ s � i=1 gi(x) + ωki(x), s � i=1 ωhi(x)⟩Fp[x], where for each 1 ≤ i ≤ s we have Zj = −Zi and gi(x) = ki(x) = hi(x) = 0 if Cj = Nj, gi(x) = hi(x) = (xn − 1)/fi(x) and ki(x) = 0 if Cj = 0, gi(x)+ωki(x) = � (xn −1)/fi(x) �� ω +ti(x) � and hi(x) = 0 if Cj = ⟨ � (xn −1)/fj(x) �� ω + sj(x) � ⟩Fp[x] and ti(x) ≡ sj(x−1) (mod fj(x)), gi(x) = (xn − 1)/fi(x) and ki(x) = hi(x) = 0 if Cj = ⟨(xn − 1)/fj(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We apply Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If Cj = Nj, then C⊥s ∩ Ni = {0} by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by the same lemma, if Cj = 0, then Ni ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves the first two bullets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 implies that ⟨ � (xn − 1)/fi(x) �� ω + ti(x) � ⟩Fp[x] ⊆ C⊥s if Cj = ⟨ � (xn − 1)/fj(x) �� ω + sj(x) � ⟩Fp[x], and ⟨(xn − 1)/fi(x)⟩Fp[x] ⊆ C⊥s if Cj = ⟨(xn − 1)/fi(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves the statements of the last two bullets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ To determine self-orthogonal and self-dual additive cyclic codes over Fp2, we need more infor- mation about irreducible factors of xn − 1 over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , Zr and Z′ 1, −Z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , Z′ t, −Z′ t be all the p-cyclotomic cosets modulo n, where Zi = −Zi and r + 2t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Each Zi is in correspondence to an irreducible polynomial fi(x) and (Z′ j, −Z′ j) are in correspondence with an irreducible pair of polynomials (fj1(x), fj2(x)) over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, we can rewrite the irreducible decomposition of xn − 1 as xn − 1 = r � i=1 fi(x) t� j=1 fj1(x)fj2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We use the above representation of cyclotomic cosets in the upcoming results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we classify self-orthogonal and self-dual additive codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s �� i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is symplectic self-orthogonal if and only if (1) for all 1 ≤ k ≤ r only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Ck = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x], where fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) for all 1 ≤ j ≤ t only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Cj1 = 0 or Cj2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First, let 1 ≤ k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, if Ck = Nk, then C⊥s ∩ Nk = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So Ck cannot have two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, if 0 ̸= Ck = ⟨((xn −1)/fk(x))(s(x)+ ω)⟩Fp[x], then ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x] ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ck is self-orthogonal if and only if Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] = ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the above equality holds if and only if fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ck is self-orthogonal if and only if one of the conditions of Part (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, let 1 ≤ j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, if Cj1 = Nj1, then C⊥s ∩ Nj2 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So if one of Cj1 or Cj2 has two generator polynomials, the other code should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the same lemma shows that if Cj1 = 0 or Cj2 = 0, then Cj1 + Cj2 is self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we concentrate only on the case when both Cj1 and Cj2 have exactly one non-zero generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, if Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x], then C⊥s ∩ Nj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, the code Cj1 ⊕ Cj2 is self-orthogonal if and only if condition (2)(b) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Condition (2)(c) follows similarly by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we use the above conditions to characterize all the symplectic self-dual additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is symplectic self-dual if and only if (1) for all 1 ≤ k ≤ r only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] where fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) for all 1 ≤ j ≤ t only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Cj1 = 0 and Cj2 = Nj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Cj2 = 0 and Cj1 = Nj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (d) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that all the self-dual additive cyclic codes over Fp2 satisfy the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 and have maximal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus the result easily follows by implying the maximal property into the conditions of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Our next goal is to compute the parameter e = dimFp(C) − dimFp(C ∩ C⊥s) for all additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameter e determines how close an additive cyclic code C is from being self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This parameter plays an important role in the quantum construction that we are applying in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let (1) B1 = {α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , αt1} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , r} such that Cαl = Nαl for all 1 ≤ l ≤ t1, (2) B2 = {β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , βt2} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , r} such that Cβl = ⟨((xn − 1)/fβl(x))(sβl(x) + ω)⟩Fp[x] and fβl ∤ sβl(x−1) − sβl(x) for all 1 ≤ l ≤ t2, POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 13 (3) B3 = {γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , γt3} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that one of Cγl1 and Cγl2 is generated by two polynomials and the other one has only one generator polynomial for all 1 ≤ l ≤ t3, (4) B4 = {κ1, κ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , κt4} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that both of Cκl1 and Cκl2 are generated by two polynomials for all 1 ≤ l ≤ t4, (5) B5 = {σ1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , σt5} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that both of Cσl1 and Cσl2 are generated by one polynomial for all 1 ≤ l ≤ t5 and (a) if Cσl1 = ⟨((xn−1)/fσl1(x))(sσl(x)+ω)⟩Fp[x], then Cσl2 ̸= ⟨((xn−1)/fσl2(x))(sσl(x−1)+ ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) if Cσl1 = ⟨(xn − 1)/fσl1(x)⟩Fp[x], then Cσl2 ̸= ⟨(xn − 1)/fσl2(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then e = dimFp(C)−dimFp(C∩C⊥s) = t1 � l=1 2|Zαl|+ t2 � l=1 |Zβl|+ t3 � l=1 2|Zγl|+ t4 � l=1 4|Zκl|+ t5 � l=1 2|Zσl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6, an additive cyclic code is not symplectic self-orthogonal if and only if at least one of the sets B1 −B5 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we consider all scenarios (1)-(5) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Let j ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ C⊥s) = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let j ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ C⊥s) = |Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) Let j ∈ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Without loss of generality we assume that Cj1 = Nj1 and Cj2 is an irreducible subcode of Nj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, the intersection C⊥s∩(Cj1⊕Cj2) is an irreducible subcode of Nj1 which implies that dimFp(Cj)−dimFp((Cj1⊕Cj2)∩C⊥s) = 3|Zj|−|Zj| = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) Let j ∈ B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 4|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (5) Let j ∈ B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In both parts (a) and (b), C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now, the result follows by combining the above observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that the case (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 never happens for Ci with deg(fi(x)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, for each 1 ≤ i ≤ r, the cyclotomic coset Zi either is a singleton or it has an even size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This is mainly because for each 0 ̸= a ∈ Zi, if a ≡ −a (mod n), then n | 2a, which implies that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence in this case p ̸= 2 (we assumed that gcd(n, p) = 1) and Zi = {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, if Zi satisfies the case (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 and |Zi| > 1, then for any a ∈ Zi, we have −a ∈ Zi and −a ̸≡ a (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that |Zi| is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This fact and the formula in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) imply that the nearly self-orthogonality parameter e of an additive cyclic code is always an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we classify additive cyclic codes with small values of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First, we need the following preliminary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let p be a prime number and gcd(n, p) = 1 for some positive number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) If gcd(n, p − 1) = d, then there are d singleton p-cyclotomic cosets modulo n and all of their coset leaders are {k n d : 0 ≤ k ≤ d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) If gcd(n, p − 1) = d and gcd(n, p2 − 1) = d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there are d′−d 2 p-cyclotomic cosets modulo n of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) The proof easily follows from the fact that {a} is a singleton coset if and only if a ≡ pa (mod n) or equivalently if and only if a(p − 1) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By elementary number theory, if gcd(n, p − 1) = d, then the latter equation has d solutions in the forms {k n d : 0 ≤ k ≤ d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) A p-cyclotomic coset modulo n containing a has size two if and only if a ≡ p2a (mod n) and a ̸≡ pa (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we get d′ candidate for the size two cosets by solving a ≡ p2a (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, each singleton cyclotomic coset is formed by a solution of the latter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note also that the p-cyclotomic coset of size two containing a and pa is counted twice in our previous observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence there are d′−d 2 many different cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ For example, for an odd n, the only singleton p-cyclotomic coset modulo n is {0} when p = 2 or p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If n is even, then {n 2 } and {0} are the only singleton cyclotomic cosets for p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The next theorem classifies all the additive cyclic codes with e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the case e = 0 happens if an additive cyclic code is symplectic self-orthogonal, and this case was characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = s � i=1 Ci be an additive cyclic code of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then e = dimFp(C) − dimFp(C ∩ C⊥s) = 2 if and only if all Ci satisfy the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 except one which is in correspondence to (1) a singleton coset and satisfies condition (1) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8, (2) a size two coset and satisfies condition (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The result follows from considering the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) and considering all conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Many of our record-breaking quantum codes provided in the next section have e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, the total number of all additive cyclic codes can be a very large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So the classification of e values significantly helps to prune the search algorithm for quantum codes with good parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' New binary quantum codes In this section, we first recall a construction of binary quantum codes from additive codes, which does not require the symplectic self-orthogonality condition of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we apply this construction to several nearly self-orthogonal additive cyclic codes over F4 and con- struct new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the rest of this section, we show the quaternary filed by F4 = {0, 1, ω, ω + 1}, where ω2 = ω + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [5, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7],[19] Let C be an (n, 2k) additive code over F4 and r = 2n − k − dimFp(C ∩ C⊥s) 2 Then there exists a binary quantum code with parameters [[n + r, k − n + r, d]]2, where d �� min{d(C), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that we take advantage of the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 in the computation of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the value of r in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 is dimFp(C⊥s)−dimFp(C∩C⊥s) 2 , where the numerator measures the nearly self-orthogonality of the code C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we briefly describe two of our new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The rest of our new binary quantum codes presented in Table 1 can be constructed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 15 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let n = 21 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 and k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is a (21, 220) additive code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, our computation using the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 shows that C has nearly self-orthogonality parameter e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, 7 = min{d(C⊥s), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So, applying the construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s gives a new quantum code with parameters [[22, 2, 7]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It has a better minimum distance than the previous best-known quantum code with the same length and dimension, which had minimum distance 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let n = 35 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 and k(x) = x34 +x33 +x31 +x30 +x29 +x27 +x25 +x23 +x22 +x20 +x19 +x18 +x15 +x12 +x8 +x3 +x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C has parameters (35, 220) as an additive cyclic code over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Also, the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 shows that C has nearly self-orthogonality parameter e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, 6 = min{d(C⊥s), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So, applying the construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s gives a record-breaking quantum code with parameters [[37, 17, 6]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The previous best-known binary quantum code with the same parameters had minimum distance 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, in order to apply the quantum construction given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1, we target additive cyclic codes with the nearly self-orthogonality e ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Because it is more likely to get a new quantum code when e value is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In Table 1, we present the parameters of our new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the table, we start with an additive cyclic code C over F4 and compute its nearly self-orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we apply the quantum construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameters of the corresponding quantum code are given in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the minimum distance of the previous quantum code with the same length and dimension is provided in the last column of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The previous minimum distance is taken from Grassl’s code table [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We record the generator polynomials of the additive cyclic codes of Table 1 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note also that applying the secondary constructions presented in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 to the new codes of Table 1 produces many more record-breaking quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the new [[52, 24, 8]]2 quantum codes alone produces the following new quantum codes: [[52, 21, 8]]2, [[52, 22, 8]]2, [[52, 23, 8]]2, [[53, 21, 8]]2, [[53, 22, 8]]2, [[53, 23, 8]]2, [[53, 24, 8]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Around the same time as us, authors of [14] independently found several new binary quantum codes by applying the connection between quasi-cyclic codes and additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, three of our new quantum codes, namely [[45, 6, 10]], [[45, 45, 10, 9]], and [[51, 8, 11]], are also among the new quantum codes of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Acknowledgement The authors would like to thank Petr Lisonˇek and Markus Grassl for many interesting dis- cussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 16 No Length e value Parameters Previous distance 1 n = 21 2 [[22, 2, 7]]2 6 2 n = 35 4 [[37, 17, 6]]2 5 3 n = 45 0 [[45, 6, 10]]2 9 4 n = 45 0 [[45, 10, 9]]2 8 5 n = 51 0 [[51, 8, 11]]2 10 6 n = 51 2 [[52, 16, 10]]2 9 7 n = 51 2 [[52, 24, 8]]2 7 8 n = 63 2 [[64, 33, 8]]2 7 9 n = 63 2 [[64, 34, 8]]2 7 10 n = 63 2 [[64, 35, 8]]2 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Parameters of new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Bierbrauer and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Edel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum twisted codes.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Foote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Abstract algebra, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Wiley Hoboken, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ezerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Good stabilizer codes from quasi-cyclic codes over F4 and F9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In 2019 IEEE International Symposium on Information Theory (ISIT), pages 2898–2902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Gottesman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Class of quantum error-correcting codes saturating the quantum Hamming bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Physical Review A, 54(3):1862, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Grassl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Code Tables: Bounds on the parameters of various types of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Symplectic self-orthogonal quasi-cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='14225, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [15] C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='ca, kh411@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='com POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 17 No Generator polynomials as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 part (II) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x34+x33+x31+x30+x29+x27+x25+x23+x22+x20+x19+x18+x15+x12+x8+x3+x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x)=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x44 + x43 + x41 + x40 + x39 + x38 + x34 + x33 + x30 + x26 + x24 + x20 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x17 + x16 + x15 + x14 + x11 + x9 + x5 + x3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x43 + x42 + x41 + x40 + x36 + x33 + x32 + x31 + x30 + x28 + x26 + x25 + x17 + x16 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x15 + x13 + x11 + x10 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x44 + x43 + x40 + x38 + x37 + x34 + x31 + x27 + x22 + x21 + x20 + x19 + x18 + x17 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x14 + x12 + x7 + x6 + x5 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x44 + x41 + x40 + x37 + x36 + x35 + x33 + x30 + x29 + x27 + x26 + x25 + x22 + x20 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x15 + x14 + x12 + x11 + x10 + x7 + x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x50 + x49 + x48 + x46 + x45 + x43 + x42 + x41 + x40 + x37 + x36 + x35 + x30 + x29 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x28 + x26 + x23 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x9 + x7 + x6 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x50 + x47 + x44 + x43 + x42 + x41 + x40 + x38 + x36 + x35 + x33 + x32 + x28 + x26 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x24 + x21 + x20 + x16 + x14 + x12 + x9 + x8 + x7 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x)=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x48 + x40 + x37 + x36 + x33 + x31 + x30 + x24 + x23 + x21 + x19 + x15 + x11 + x10 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x9 + x8 + x7 + x4 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x41 + x40 + x36 + x35 + x34 + x33 + x30 + x29 + x27 + x23 + x22 + x21 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x16 + x13 + x12 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x50 + x49 + x48 + x47 + x46 + x45 + x44 + x41 + x40 + x39 + x33 + x31 + x30 + x28 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x25 + x22 + x20 + x19 + x18 + x17 + x16 + x14 + x13 + x11 + x9 + x5 + x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x49 + x48 + x46 + x44 + x43 + x41 + x38 + x37 + x36 + x33 + x32 + x31 + x30 + x29 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x27 + x25 + x21 + x20 + x18 + x17 + x15 + x11 + x10 + x7 + x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x43 + x42 + x41 + x40 + x38 + x37 + x33 + x32 + x30 + x26 + x24 + x22 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x16 + x15 + x13 + x9 + x5 + x4 + x2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x50 +x49 +x48 +x47 +x46 +x45 +x44 +x43 +x42 +x41 +x40 +x39 +x38 +x37 +x36 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x35 +x34 +x33 +x32 +x31 +x30 +x29 +x28 +x27 +x26 +x25 +x24 +x23 +x22 +x21 +x20 +x19 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x18 +x17 +x16 +x15 +x14 +x13 +x12 +x11 +x10 +x9 +x8 +x7 +x6 +x5 +x4 +x3 +x2 +x+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x61+x60+x59+x57+x56+x53+x52+x51+x42+x41+x38+x36+x34+x32+x31+x28+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x27 +x26 +x24+x20 +x19+x16 +x14+x13 +x12+x11 +x9+x8+x7+x6+x5 +x4+x3+x2+x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x61 + x59 + x57 + x56 + x55 + x54 + x52 + x51 + x50 + x49 + x47 + x44 + x37 + x36 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x35 + x33 + x32 + x31 + x29 + x28 + x27 + x26 + x24 + x22 + x21 + x16 + x8 + x5 + x3 + x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x60 + x59 + x58 + x55 + x54 + x53 + x52 + x51 + x48 + x47 + x45 + x44 + x40 + x38 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x37 + x36 + x35 + x34 + x33 + x32 + x31 + x30 + x29 + x28 + x27 + x24 + x23 + x22 + x21 + x15 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x13 + x10 + x9 + x7 + x6 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x62 + x59 + x56 + x55 + x54 + x53 + x49 + x47 + x46 + x42 + x41 + x40 + x37 + x35 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x33 + x31 + x29 + x28 + x27 + x24 + x20 + x19 + x16 + x15 + x14 + x7 + x4 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x61+x60+x59+x58+x57+x53+x52+x49+x44+x41+x38+x37+x36+x35+x34+x32+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x31 +x30+x27+x26+x24+x23+x21+x20 +x19+x13+x12+x11+x8+x6+x5+x4+x3+x+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x60+x58+x57+x56+x52+x48+x47+x46+x44+x42+x40+x39+x38+x36+x35+x34+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x32+x31+x30+x26+x25+x24+x22+x19+x18+x17+x13+x12+x9+x7+x6+x5+x4+x3+x2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Generator polynomials of additive cyclic codes of Table 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'}