# Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes

Hao Chen, Zhiqiang Cheng and Yanfeng Qi Hao Chen and Zhiqiang Cheng are with
the College of Information Science and Technology/Cyber Security, Jinan
University, Guangzhou, Guangdong Province, 510632, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Yanfeng Qi is with School of Science, Hangzhou Dianzi
University, Hangzhou, 310018, China<EMAIL_ADDRESS>The research of Hao
Chen was supported by NSFC Grant 62032009. The research of Yanfeng Qi was
supported by the Zhejiang provincial Natural Science Foundation of China (No.
LY21A010013) and Scientific Research Fund of Zhejiang Provincial Education
Department (No.Y202249655).

###### Abstract

Sum-rank-metric codes have wide applications in the multishot network coding
and the distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes
and their Welch-Berlekamp type decoding algorithms were proposed and studied.
They are sum-rank versions of Reed-Solomon codes and BCH codes in the Hamming
metric. In this paper, we construct binary linear sum-rank-metric codes of the
matrix size $2\times 2$, from BCH, Goppa and additive quaternary Hamming
metric codes. Larger sum-rank-metric codes than these sum-rank BCH codes of
the same minimum sum-rank distances are obtained. Then a reduction of the
decoding in the sum-rank-metric to the decoding in the Hamming metric is
given. Fast decoding algorithms of BCH and Goppa type binary linear sum-rank-
metric codes of the block length $t$ and the matrix size $2\times 2$, which
are better than these sum-rank BCH codes, are presented. These fast decoding
algorithms for BCH and Goppa type binary linear sum-rank-metric codes of the
matrix size $2\times 2$ need at most $O(t^{2})$ operations in the field ${\bf
F}_{4}$. Asymptotically good sequences of quadratic-time encodable and
decodable binary linear sum-rank-metric codes of the matrix size $2\times 2$
satisfying

$R_{sr}(\delta_{sr})\geq
1-\frac{1}{2}(H_{4}(\frac{4}{3}\delta_{sr})+H_{4}(2\delta_{sr})),$

can be constructed from Goppa codes.

Index terms: Sum-rank BCH code, Decoding of sum-rank-metric code, Welch-
Berlekamp decoding algorithm, Additive quaternary code, Quadratic-time
encodable and decodable sum-rank-metric codes.

##  1 Introduction

Sum-rank-metric codes have wide applications in multishot network coding, see
[30, 36, 40], space-time coding, see [43], and coding for distributed storage,
see [11, 31, 28]. There have been many papers on structures, constructions and
applications of sum-rank-metric codes in recent years, see e.g.[10, 9, 28, 29,
30, 33, 32, 38, 39, 35] and references therein. Fundamental properties and
some bounds on sizes of sum-rank-metric codes were given in the paper [9]. For
a nice survey of sum-rank-metric codes and their applications, we refer to
[34].

We recall some basic concepts and results for sum-rank-metric codes in [9].
Let ${\bf F}_{q}^{(n,m)}$ be the set of all $n\times m$ matrices. This is a
linear space over ${\bf F}_{q}$ of the dimension $nm$. Let $n_{i}\leq m_{i}$
be $2t$ positive integers satisfying $m_{1}\geq m_{2}\cdots\geq m_{t}$. Set
$N=n_{1}+\cdots+n_{t}$. Let

${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}={\bf F}_{q}^{n_{1}\times
m_{1}}\bigoplus\cdots\bigoplus{\bf F}_{q}^{n_{t}\times m_{t}}$

be the set of all ${\bf x}=({\bf x}_{1},\ldots,{\bf x}_{t})$, ${\bf
x}_{i}\in{\bf F}_{q}^{n_{i}\times m_{i}}$, $i=1,\ldots,t$. This is a linear
space over ${\bf F}_{q}$ of the dimension $\Sigma_{i=1}^{t}n_{i}m_{i}$. We
call $n_{1}\times m_{1},\ldots,n_{t}\times m_{t}$, the matrix sizes of sum-
rank-metric codes. Set $wt_{sr}({\bf x}_{1},\ldots,{\bf x}_{t})=rank({\bf
x}_{1})+\cdots+rank({\bf x}_{t})$ and

$d_{sr}({\bf x},{\bf y})=wt_{sr}({\bf x}-{\bf y}),$

for ${\bf x},{\bf y}\in{\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. This
is indeed a metric on ${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$.

Definition 1.1. A block length $t$ and matrix sizes $n_{1}\times
m_{1},\ldots,n_{t}\times m_{t}$ sum-rank-metric code ${\bf C}\subset{\bf
F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$ is a subset of the finite metric
space ${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. Its minimum sum-rank
distance is defined by

$d_{sr}({\bf C})=\min_{{\bf x}\neq{\bf y},{\bf x},{\bf y}\in{\bf
C}}d_{sr}({\bf x},{\bf y}).$

The code rate of ${\bf C}$ is $R_{sr}=\frac{log_{q}|{\bf
C}|}{\Sigma_{i=1}^{t}n_{i}m_{i}}$. The relative distance is
$\delta_{sr}=\frac{d_{sr}}{N}$. When ${\bf C}\subset{\bf
F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$ is a linear subspace of the
linear space ${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$ over ${\bf
F}_{q}$, ${\bf C}$ is called a linear sum-rank-metric code. When $q=2$, this
code is called binary sum-rank-metric code.

In general it is interesting to construct good sum-rank-metric codes with
large cardinalities and large minimum sum-rank distances, and develop their
decoding algorithms. For $t=1$, this is the rank-metric code case, see [16].
For $m_{1}=\cdots=m_{t}=m$, $n_{1}=\cdots=n_{t}=n$, this is the $t$-sum-rank-
metric code over ${\bf F}_{q^{m}}$ with the length $nt$. For $m=n=1$, this is
the Hamming metric code case. Hence the sum-rank-metric is a generalization
and combination of the Hamming metric and the rank-metric. When $q=2$,
$n_{1}=\cdots=n_{t}=m_{1}=\cdots=m_{t}=2$, sum-rank-metric codes are in the
space ${\bf F}_{2}^{(2,2),\ldots,(2,2)}={\bf F}_{2}^{2\times
2}\bigoplus\cdots\bigoplus{\bf F}_{2}^{2\times 2}$, this is the case combining
binary Hamming metric codes and $2\times 2$ binary matrices. In this paper we
concentrate on the construction and decoding of binary linear sum-rank-metric
codes in ${\bf F}_{2}^{(2,2),\ldots,(2,2)}$.

When the matrix size $n\times m$ is fixed, a sequence of sum-rank-metric codes
${\bf C}_{i}$ of the block size $t_{i}\longrightarrow\infty$, satisfying

$R_{sr}=\lim\frac{log_{q}|{\bf C}_{i}|}{mnt_{i}}>0$

and

$\delta_{sr}=\lim\frac{d_{sr}({\bf C}_{i})}{nt_{i}}>0,$

is called an asymptotically good sequence of sum-rank-metric codes. It is
interesting to get asymptotic bound on $R_{sr}(\delta_{sr})$.

The Hamming weight $wt_{H}({\bf a})$ of a vector ${\bf
a}=(a_{1},\ldots,a_{n})\in{\bf F}_{q}^{n}$ is the number of coordinate
positions in its support

$supp({\bf a})=\\{i:a_{i}\neq 0\\}.$

The Hamming distance $d_{H}({\bf a},{\bf b})$ between two vectors ${\bf a}$
and ${\bf b}$ is defined to be the Hamming weight $wt_{H}({\bf a}-{\bf b})$.
For a code ${\bf C}\subset{\bf F}_{q}^{n}$ of dimension $k$, its Hamming
distance is

$d_{H}=\min_{{\bf a}\neq{\bf b}}\\{d_{H}({\bf a},{\bf b}):{\bf a},{\bf
b}\in{\bf C}\\}.$

It is well-known that the Hamming distance of a linear code ${\bf C}$ is the
minimum Hamming weight of its non-zero codewords. For the theory of error-
correcting codes in the Hamming metric, we refer to the books [27, 25, 23]. A
code ${\bf C}$ in ${\bf F}_{q}^{n}$ of the minimum distance $d_{H}$ and the
cardinality $M$ is denoted by an $(n,M,d_{H})_{q}$ code. The code rate is

$R({\bf C})=\frac{log_{q}M}{n},$

and the relative distance is

$\delta({\bf C})=\frac{d_{H}}{n}.$

For a linear $[n,k,d_{H}]_{q}$ code, the Singleton bound asserts $d_{H}\leq
n-k+1$. When equality holds, this code is called a maximal distance separable
(MDS) code and Reed-Solomon codes are well-known MDS codes, see e.g.[23]. BCH
codes and Goppa codes can be considered as subfield subcodes of generalized
Reed-Solomon codes, and are severed as optimal binary codes for many
parameters, see [27, 23, 19].

Each Hamming metric $[t,k,d_{H}]_{q^{n}}$ code can be considered as a sum-
rank-metric code of the matrix size $n\times n$ and the block length $t$ over
${\bf F}_{q}$, by taking each coordinate in ${\bf F}_{q^{n}}$ as the first row
of the $n\times n$ matrix, and padding zero other rows to this $n\times n$
matrix. This is a dimension $nk$ sum-rank-metric code, with the minimum sum-
rank distance at least $d_{H}$. Then the code rate of this sum-rank-metric
code is

$R_{sr}=\frac{nk}{n^{2}t}=\frac{R({\bf C})}{n}$

and

$\delta_{sr}=\frac{d_{H}}{nt}=\frac{\delta({\bf C})}{n}.$

Each Hamming metric $[nt,k,d_{H}]_{q^{n}}$ code can be considered as a sum-
rank-metric code of the matrix size $n\times n$ and the block length $t$ over
${\bf F}_{q}$. By taking every consecutive $n$ coordinates in ${\bf
F}_{q^{n}}$ as the $n\times n$ matrix. This is a dimension $nk$ sum-rank-
metric code, with the minimum sum-rank distance at least $\frac{d_{H}}{n}$.
Then the code rate of this sum-rank-metric code is

$R_{sr}=\frac{nk}{n^{2}t}=R({\bf C})$

and

$\delta_{sr}\geq\frac{d_{H}}{n^{2}t}=\frac{\delta({\bf C})}{n}.$

Therefore asymptotically good sequences of sum-rank-metric codes of the matrix
size $n\times n$ can be obtained directly from asymptotically good sequence of
Hamming metric codes meeting the Gilbert-Varshamov bound. The code rate and
the relative distance of this sequence of sum-rank-metric codes satisfy

$R_{sr}\geq 1-H_{q^{n}}(n\delta_{sr}),$

where $H_{q^{n}}(x)$ is the $q^{n}$-ary entropy function. The $q$-ary entropy
function is defined on $0\leq x\leq 1-\frac{1}{q}$ by

$H_{q}(0)=0$

and

$H_{q}(x)=xlog_{q}(q-1)-xlog_{q}x-(1-x)log_{q}(1-x),$

for $x\neq 0$. Asymptotically good sequences of quadratic-time encodable and
decodable sum-rank-metric codes obtained in Section 4 have much better
asymptotic parameters than the above bound.

The rank-metric on the space ${\bf F}_{q}^{(m,n)}$ of size $n\times m$
matrices over ${\bf F}_{q}$, where $n\leq m$, is defined by the ranks of
matrices, $d_{r}(A,B)=rank(A-B)$. The minimum rank-distance of a code ${\bf
C}\subset{\bf F}_{q}^{(m,n)}$ is

$d_{r}({\bf C})=\min_{A\neq B}\\{d_{r}(A,B):A,B\in{\bf C}\\}.$

For a code ${\bf C}$ in ${\bf F}_{q}^{(m,n)}$ with the minimum rank distance
$d_{r}({\bf C})\geq d$, the Singleton-like bound asserts that the number of
codewords in ${\bf C}$ is upper bounded by $q^{m(n-d+1)}$ , see [16]. A code
satisfying the equality is called a maximal rank distance (MRD) code. The
Gabidulin code in ${\bf F}_{q}^{(n,n)}$ is consisting of ${\bf F}_{q}$ linear
mappings on ${\bf F}_{q}^{n}\cong{\bf F}_{q^{n}}$ defined by $q$-polynomials
$a_{0}x+a_{1}x^{q}+\cdots+a_{i}x^{q^{i}}+\cdots+a_{t}x^{q^{t}}$, where
$a_{t},\ldots,a_{0}\in{\bf F}_{q^{n}}$ are arbitrary elements in ${\bf
F}_{q^{n}}$, see [16]. Then the rank-distance of the Gabidulin code is at
least $n-t$ since each above $q$-polynomial has at most $q^{t}$ roots in ${\bf
F}_{q^{n}}$. There are $q^{n(t+1)}$ such $q$-polynomials. Hence the size of
the Gabidulin code is $q^{n(t+1)}$ and it is an MRD code.

The Singleton-like bound for the sum-rank-metric was proposed in [31, 9]. The
general form Theorem III.2 in [9] is as follows. The minimum sum-rank distance
$d$ can be written uniquely as the form
$d_{sr}=\Sigma_{i=1}^{j-1}n_{i}+\delta+1$ where $0\leq\delta\leq n_{j}-1$,
then

$|{\bf C}|\leq q^{\Sigma_{i=j}^{t}n_{i}m_{i}-m_{j}\delta}.$

The code attaining this bound is called a maximal sum-rank-metric distance
(MSRD) code. When $m_{1}=\cdots=m_{t}=m$ this bound is of the form

$|{\bf C}|\leq q^{m(N-d_{sr}+1)}.$

For constructions of some MSRD codes, we refer to [9, 33, 14]. Linearized
Reed-Solomon codes, which are the sum-rank versions of the classical Reed-
Solomon codes, were introduced and studied in [28]. Their Welch-Berlekamp
decoding algorithm was presented in [30].

There have been few constructions of sum-rank-metric code with good
parameters, we refer to the subfield subcodes of linearized Reed-Solomon codes
in [28] and sum-rank BCH codes in [32]. Sum-rank BCH codes of the matrix size
$n_{1}=\cdots=n_{t}=n$, $m_{1}=\cdots=m_{t}=m$ were proposed and studied in
[32] by the deep algebraic method. These sum-rank-metric codes are linear over
${\bf F}_{q^{m}}$. There is a designed distance of such a sum-rank BCH code
such that the minimum sum-rank distance is greater than or equal to the
designed distance. On the other hand the dimensions of these sum-rank BCH
codes are lower bounded in [32] Theorem 9. Many sum-rank-metric codes of the
parameters $n=m=2$ and $q=2$ were constructed in Tables I, II, III, IV, V, VI
and VII of [32]. The minimum sum-rank distances of these sum-rank BCH codes
were improved in a recent paper [1]. The decoding algorithm for sum-rank BCH
codes was presented in [32]. In our previous paper [14] we proposed a new
construction of linear sum-rank-metric codes by combining several Hamming
metric codes and $q$-polynomial representations of rank-metric codes. Most of
constructed sum-rank-metric codes in the paper [14] from presently known best
linear codes in [19] are larger than the sum-rank BCH codes in [32] of the
same minimum sun-rank distances. No decoding algorithm has been developed for
sum-rank-metric codes in [14].

Fast algebraic decoding and list-decoding algorithms for BCH codes, Goppa
codes and algebraic geometry codes in the Hamming metric are well-developed,
see [23, Chapter 5], [15, 13] and [20]. It is well-known that BCH codes and
Goppa codes over ${\bf F}_{q}$ in the Hamming metric can be decoded up to
$\left\lfloor\frac{d-1}{2}\right\rfloor$ errors within the cost of $O(t^{2})$
operations of the field ${\bf F}_{q}$, where $t$ is the length of the code, we
refer to [25, Chapter 6], [23, Chapter 5] and [27, Chapter 9, Chapter 12]. To
the best of our knowledge, there are few known decoding algorithms of sum-
rank-metric codes. The fundamental decoding algorithms in the sum-rank-metric
are the Welch-Berlekamp decoding algorithm for linearized Reed-Solomon codes
given in [30] and the decoding algorithm for the sum-rank BCH codes given in
[32]. The complexity of their decoding algorithms is $O(t^{2})$ operations in
the field ${\bf F}_{q^{m}}$. For the fast decodings of linearized Reed-Solomon
codes, interleaved linearized Reed-Solomon codes and folded linearized Reed-
Solomon codes, we refer to [2, 22, 24, 3]. Generic decoder in the sum-rank-
metric was developed in [41].

In this paper we construct linear sum-rank-metric codes over ${\bf F}_{2}$ of
the matrix size $2\times 2$ from quaternary BCH and Goppa codes. These binary
linear sum-rank-metric codes are called BCH and Goppa type sum-rank-metric
codes. Many BCH and Goppa type binary linear sum-rank-metric codes are better
than the sum-rank BCH codes constructed by U. Martínez-Peñas in [28, 32].
Additive quaternary codes were studied from the motivation of finite
geometries and the construction of quantum codes in [7, 5, 4, 6]. We indicate
that these additive quaternary codes can be used to construct better binary
linear sum-rank-metric codes of the matrix size $2\times 2$. Importantly some
good explicit binary linear sum-rank-metric codes constructed in [14] can be
improved from these additive quaternary codes.

We show that how to reduce the decoding of BCH and Goppa type binary linear
sum-rank-metric codes to the decoding of quaternary Hamming metric BCH and
Goppa codes. Then from various fast decoding algorithms of quaternary BCH and
Goppa codes, we present fast decoding algorithms of our BCH and Goppa type
binary linear sum-rank-metric codes. The complexity is $O(t^{2})$ operations
in the field ${\bf F}_{q^{2}}={\bf F}_{4}$. Hence our decoding algorithm has
the same complexity as the decoding algorithm presented in [32]. However our
BCH type sum-rank-metric codes are larger than these sum-rank BCH codes of the
same minimum sum-rank distances. Our reduction of decoding is general and then
can be applied to binary linear sum-rank-metric codes of the matrix size
$2\times 2$ constructed from any quaternary Hamming metric codes with the fast
decoding. Asymptotically good sequences of binary linear sum-rank-metric codes
of the matrix size $2\times 2$ are constructed from Goppa codes, which can be
decoded within the $O(t^{2})$ operations of the field ${\bf F}_{4}$.

Two of the main results of this paper are as follows.

Main result 1. By the usage of two quaternary BCH codes, explicit block length
$4^{h}-1$, $h=2,3,\ldots$, and matrix size $2\times 2$ binary linear sum-rank-
metric codes, which are larger than the sum-rank BCH codes of the same minimum
sum-rank distances, can be constructed. These BCH type binary linear sum-rank-
metric codes can be decoded up to
$\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors, within the complexity of
$O(t^{2})$ operations in the field ${\bf F}_{4}$.

Main result 2. By the usage of two quaternary Goppa codes, asymptotically good
sequence of binary linear sum-rank-metric codes of the matrix size $2\times 2$
satisfying

$R_{sr}(\delta_{sr})\geq
1-\frac{1}{2}(H_{4}(\frac{4}{3}\delta_{sr})+H_{4}(2\delta_{sr})),$

which can be efficiently encoded and decoded within $O(t^{2})$ operations of
the field ${\bf F}_{4}$, are constructed.

The class of linear sum-rank-metric codes of the matrix size $2\times 2$ is
the first natural extension of linear error-correcting codes in the Hamming
metric, which can be considered as linear sum-rank-metric codes of the matrix
size $1\times 1$. In this paper we concentrate on the construction and
decoding of linear sum-rank-metric codes of the matrix size $2\times 2$. In
the matrix size $1\times 1$ case, the main results in [44] gave the
construction of asymptotically good sequence of linear-time encodable and
decodable error-correcting codes.

##  2 BCH and Goppa type sum-rank-metric codes

In this section, we construct linear sum-rank-metric codes for the parameters
$q=2$ and $n_{1}=\cdots=n_{t}=m_{1}=\cdots=m_{t}=2$. First of all the linear
space ${\bf F}_{q}^{(n,n)}$ of all $n\times n$ matrices over ${\bf F}_{q}$ can
be identified with the space of all $q$-polynomials,

${\bf
F}_{q}^{(n,n)}=\\{a_{0}x+a_{1}x^{q}+\cdots+a_{n-1}x^{q^{n-1}}:a_{0},\ldots,a_{n-1}\in{\bf
F}_{q^{n}}\\}.$

This is the isomorphism of the linear space over ${\bf F}_{q}$.

The construction of binary linear sum-rank-metric codes of the matrix size
$2\times 2$ from two linear quaternary codes is as follows. The first linear
$[t,k_{1},d_{1}]_{4}$ code ${\bf C}_{1}\subset{\bf F}_{4}^{t}$ corresponds to
the coefficients of $x^{2}$ in the $2$-polynomial representations of the
$2\times 2$ matrix. The second linear $[t,k_{2},d_{2}]_{4}$ code ${\bf
C}_{2}\subset{\bf F}_{4}^{t}$ corresponds to the coefficients of $x$ in the
$2$-polynomial representations of the $2\times 2$ matrix. Then

$SR({\bf C}_{1},{\bf C}_{2})=\\{{\bf a}_{2}x+{\bf a}_{1}x^{2}:{\bf
a}_{2}=(a_{1,2},\ldots,a_{t,2})\in{\bf C}_{2},{\bf
a}_{1}=(a_{1,1},\ldots,a_{t,1})\in{\bf C}_{1}\\}.$

This is a linear (over ${\bf F}_{2}$) sum-rank-metric code with the minimum
sum-rank distance at least $\min\\{d_{1},2d_{2}\\}$. It is easy to verify the
linear independence, therefore the dimension $\dim_{{\bf F}_{2}}(SR({\bf
C}_{1},{\bf C}_{2}))=2(k_{1}+k_{2})$. We have the following result.

Theorem 2.1. Let ${\bf C}_{1}\subset{\bf F}_{4}^{t}$ and ${\bf
C}_{2}\subset{\bf F}_{4}^{t}$ be two linear $[t,k_{1},d_{1}]_{4}$ and
$[t,k_{2},d_{2}]_{4}$ codes over ${\bf F}_{4}$. Then a block length $t$ and
matrix size $2\times 2$ binary linear sum-rank-metric code $SR({\bf
C}_{1},{\bf C}_{2})$ of the minimum sum-rank distance at least
$\min\\{d_{1},2d_{2}\\}$ can be constructed explicitly. The dimension of
$SR({\bf C}_{1},{\bf C}_{2})$ over ${\bf F}_{2}$ is $\dim_{{\bf
F}_{2}}(SR({\bf C}_{1},{\bf C}_{2}))=2(k_{1}+k_{2})$.

Proof. It is clear that the dimension of the linear sum-rank-metric code
$SR({\bf C}_{1},{\bf C}_{2})$ over ${\bf F}_{2}$ is $2(k_{1}+k_{2})$. For any
codeword ${\bf c}={\bf a}_{2}x+{\bf a}_{1}x^{2}\in SR({\bf C}_{1},{\bf
C}_{2})$, if ${\bf a}_{1}\in{\bf C}_{1}$ is nonzero, there are at least
$d_{1}$ positions such that the $2\times 2$ coordinate matrix is not zero.
Then $wt_{sr}({\bf c})\geq d_{1}$. Otherwise if ${\bf a}_{1}$ is a zero
codeword, then ${\bf c}={\bf a}_{2}x$, then $wt_{sr}({\bf c})=2wt_{H}({\bf
a}_{2})\geq 2d_{2}$. The conclusion is proved.

The above construction was given in our previous paper [14]. Actually this
construction can be extended as follows.

Theorem 2.2. Let ${\bf C}_{1}\subset{\bf F}_{4}^{t}$ and ${\bf
C}_{2}\subset{\bf F}_{4}^{t}$ be two linear $[t,k_{1},d_{1}]_{4}$ and
$[t,k_{2},d_{2}]_{4}$ codes over ${\bf F}_{4}$. Then a block length $t$ and
matrix size $2\times 2$ binary linear sum-rank-metric code $SR({\bf
C}_{1},{\bf C}_{2})$ can be constructed explicitly. The minimum sum-rank
distance of $SR({\bf C}_{1},{\bf C}_{2})$ is at least
$\min\\{d_{2},2d_{1}\\}$. The dimension of $SR({\bf C}_{1},{\bf C}_{2})$ over
${\bf F}_{2}$ is $\dim_{{\bf F}_{2}}(SR({\bf C}_{1},{\bf
C}_{2}))=2(k_{1}+k_{2})$. Moreover for any two codewords, ${\bf a}_{1}\in{\bf
C}_{1}$ and ${\bf a}_{2}\in{\bf C}_{2}$, set

$I=supp({\bf a}_{1})\cap supp({\bf a}_{2}),$

then

$wt_{sr}({\bf a}_{2}x+{\bf a}_{1}x^{2})=2wt_{H}({\bf a}_{1})+2wt_{H}({\bf
a}_{2})-3|I|.$

Proof. For the coordinate position $i\in supp({\bf a}_{1})\backslash supp({\bf
a}_{2})$, it is clear the matrix ${\bf a}_{2}x+{\bf a}_{1}x^{2}$ in this
position is of the rank $2$. For the coordinate position $i\in supp({\bf
a}_{2})\backslash supp({\bf a}_{1})$, it is clear the matrix ${\bf
a}_{2}x+{\bf a}_{1}x^{2}$ in this position is of the rank $2$. For these
coordinate positions $i\in I$, it is clear that the matrix is of rank $1$,
since there is one root $x=\frac{a_{i,1}}{a_{i,2}}$, where

${\bf a}_{1}=(a_{1,1},\ldots,a_{i,1}),$ ${\bf
a}_{2}=(a_{1,2},\ldots,a_{i,2}).$

Then it follows that

$wt_{sr}({\bf a}_{2}x+{\bf a}_{1}x^{2})=2(wt_{H}({\bf
a}_{1})-|I|)+2(wt_{H}({\bf a}_{1})-|I|)+|I|.$

The conclusion follows immediately.

From Theorem 2.2, two linear quaternary codes ${\bf C}_{1}$ and ${\bf C}_{2}$
can exchange their positions. It seems that two resulted binary linear sum-
rank-metric codes are not equivalent.

###  2.1 BCH type sum-rank-metric codes

When both ${\bf C}_{1}$ and ${\bf C}_{2}$ are BCH codes, we call the
constructed binary linear sum-rank-metric codes BCH type sum-rank-metric
codes, for distinguishing from these sum-rank BCH codes in [32]. Actually,
when both ${\bf C}_{1}$ and ${\bf C}_{2}$ are cyclic code, then for each
codeword $(A_{1},\ldots,A_{t})\in{\bf F}_{2}^{(2,2,),\ldots,(2,2)}$ in the
constructed sum-rank-metric code in Theorem 2.1, where $A_{i}$,
$i=1,2,\ldots,t$, is a $2\times 2$ matrix over ${\bf F}_{2}$,
$(A_{t},A_{1},\ldots,A_{t-1})$ is clearly another codeword. By the comparison
with Definition 4 in [32], this satisfies one requirement of cyclic-skew-
cyclic (CSC) sum-rank-metric code in [32]. It seems that the requirement of
CSC sum-rank-metric codes are stronger and this is the reason that our sum-
rank-metric codes are larger than these CSC codes with same minimum sum-rank
distances in [32].

Notice that there are several possibilities to take binary or quaternary BCH
codes in Theorem 2.1. We can apply Theorem 2.1 to two binary BCH codes, two
quaternary BCH codes, or one binary BCH code and one quaternary BCH code. It
is interesting to observe that the resulted linear sum-rank-metric codes from
quaternary BCH codes are much larger than sum-rank BCH codes of the same
minimum sum-rank distances invented by Martínez-Peñas in [32]. Sometimes even
our constructions in Theorems 2.1 and 2.2 are applied to two binary BCH codes,
the resulted linear sum-rank-metric codes are larger than sum-rank BCH codes
of the same minimum sum-rank distances.

Example 2.1. Let ${\bf C}_{2}$ be the length $63$ primitive narrow-sense
binary BCH code with the defining set $\\{0\\}$, $\\{1,2,4,8,16,32\\}$,
$\\{3,6,12,24,48,33\\}$, $\\{5,10,20,40,17,34\\}$. This is a binary
$[63,44,7]_{2}$ code. Another binary BCH code ${\bf C}_{1}$ with the defining
set $\\{0\\}$, $\\{1,2,4,8,16,32\\}$, $\\{3,6,12,24,48,33\\}$,
$\\{5,10,20,40,17,34\\}$, $\\{7,14,28,56,49,35\\}$, $\\{9,18,36\\}$,
$\\{11,22,44,25,50,37\\}$. This is a binary $[63,29,14]_{2}$ code. These two
binary codes can be considered as linear codes over ${\bf F}_{4}$. From
Theorem 2.1, a block length $63$ and matrix size $2\times 2$ binary linear
sum-rank-metric code of the minimum sum-rank distance at least $14$ can be
constructed. The dimension of this code is $2\cdot(29+44)=2\cdot 73$. This is
larger than the dimension $2\cdot 70$ sum-rank BCH code of the same minimum
sum-rank distance $14$ in [32, Table VI].

Example 2.2. We can use two BCH codes over ${\bf F}_{4}$. Let ${\bf C}_{2}$ be
the length $63$ primitive narrow-sense BCH code over ${\bf F}_{4}$ with the
defining set $\\{0\\}$, $\\{1,4,16\\}$, $\\{2,8,32\\}$, $\\{3,12,48\\}$,
$\\{5,20,17\\}$. This is a binary $[63,50,7]_{4}$ code ${\bf C}_{2}$. Another
BCH code ${\bf C}_{1}$ over ${\bf F}_{4}$ with the defining set $\\{0\\}$,
$\\{1,4,16\\}$, $\\{2,8,32\\}$, $\\{3,12,48\\}$, $\\{5,20,17\\}$,
$\\{6,24,33\\}$, $\\{7,28,49\\}$, $\\{9,36,18\\}$, $\\{10,40,34\\}$,
$\\{11,44,50\\}$. This is a binary $[63,35,14]_{4}$ code. From Theorem 2.1, a
block length $63$ and matrix size $2\times 2$ binary linear sum-rank-metric
code of the minimum sum-rank distance at least $14$ is constructed. The
dimension of this code is $2\cdot(50+35)=2\cdot 85$. This is much larger than
the dimension $2\cdot 70$ sum-rank BCH code of the same minimum sum-rank
distance $14$ in [32, Table VI]. It is also better than linear sum-rank-metric
codes of the dimension $2\cdot 73$ constructed in Example 3.1. Therefore it is
always good to use two BCH codes over ${\bf F}_{4}$ to construct linear sum-
rank-metric codes in Theorem 2.1.

Example 2.3. We can use one binary BCH code and one BCH code over ${\bf
F}_{4}$. Let ${\bf C}_{2}$ be the length $63$ binary BCH $[63,44,7]_{2}$ code.
Another BCH code ${\bf C}_{1}$ is the $[63,35,14]_{4}$ code over ${\bf
F}_{4}$. From Theorem 2.1, a block length $63$ and matrix size $2\times 2$
binary linear sum-rank-metric code of the minimum sum-rank distance at least
$14$ is constructed. The dimension of this code is $2\cdot(44+35)=2\cdot 79$.
This is larger than the dimension $2\cdot 70$ sum-rank BCH code of the same
minimum sum-rank distance $14$ in [32, Table VI]. It is also larger than
linear sum-rank-metric codes of the dimension $2\cdot 73$ constructed in
Example 2.2.

Corollary 2.1. A block length $4^{m}-1$ and matrix size $2\times 2$ binary
linear sum-rank-metric code can be constructed explicitly from two BCH
quaternary codes ${\bf C}_{1}$ and ${\bf C}_{2}$ of the minimum distances
$d_{1}$ and $d_{2}$ satisfying $d_{1}=2d_{2}$. The minimum sum-rank distance
of this sum-rank-metric code at least $2d_{2}=d_{1}$. The dimension of this
sum-rank-metric code is at least

$2(2\cdot 4^{m}-2-m(\frac{3d_{1}}{2}-2)).$

Proof. It is well-known that the BCH code of the minimum distance $d_{1}$ has
its dimension at least $4^{m}-1-m(d_{1}-1)$, see [23, Theorem 5.1.7]. The
conclusion follows directly.

However the dimension in Corollary 2.1 is only a lower bound. Sometimes the
explicit dimension calculation from cyclotomic cosets is better.

Example 2.4. For $m=2$, $d_{2}=3$, a block length $15$ and matrix size
$2\times 2$ binary linear sum-rank-metric code of the minimum sum-rank
distance at least $6$ is constructed from $[15,8,6]_{4}$ and $[15,12,3]_{4}$
BCH codes. The dimension of this code is $2\cdot 20$. From Corollary 2.1, the
lower bound of the dimension of this sum-rank-metric code is at least $2\times
16$, since the lower bound on dimensions of BCH codes in Corollary 2.1 is not
explicit. This linear sum-rank-metric code is larger than the constructed sum-
rank-metric code of the same minimum sum-rank distance $6$, the dimension
$2\cdot 16$ in [32, Table IV].

For $d_{2}=7$, we have two $[15,7,7]_{4}$ and $[15,1,14]_{4}$ BCH codes over
${\bf F}_{4}$. We get a block length $15$ and matrix size $2\times 2$ binary
linear sum-rank-metric code of the minimum sum-rank distance at least $14$.
The dimension of this code is $2\cdot 8$. Our code is much larger than the
sum-rank BCH code of the dimension $2\cdot 2$ and the minimum sum-rank
distance $14$, constructed in [32, Table IV].

In the following tables, we list all binary linear sum-rank-metric codes of
the block length $15$, block length $31$, block length $63$, block length
$127$ and block length $255$ constructed from two BCH codes over ${\bf
F}_{4}$. The Singleton-like bounds are also listed. Some of our codes from two
BCH quaternary codes are close to the Singleton-like bound. Many of them are
larger than sum-metric-codes of the same minimum sum-rank distances
constructed in [32]. In Tables of [32], the dimensions of sum-rank-metric
codes were only calculated for some minimum sum-rank distances. We calculated
the dimensions of sum-rank-metric codes from Theorem 9 in [32] for other
minimum sum-rank distances. However it should be indicated that the true
dimensions and the true minimum sum-rank distances of sum-rank BCH codes were
not calculated explicitly in [32]. The entries from [32] are lower bounds of
dimensions and distances.

Table 1:  Block length $t=15$, $n=m=2$. $d_{sr}$ | dimension | Table IV, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 24$ | $2\cdot 20$ | $2\cdot 27$
$5$ | $2\cdot 21$ | $2\cdot 18$ | $2\cdot 26$
$6$ | $2\cdot 20$ | $2\cdot 16$ | $2\cdot 25$
$7$ | $2\cdot 17$ | $2\cdot 14$ | $2\cdot 24$
$8$ | $2\cdot 15$ | $2\cdot 10$ | $2\cdot 23$
$9$ | $2\cdot 13$ | $2\cdot 8$ | $2\cdot 22$
$10$ | $2\cdot 13$ | $2\cdot 8$ | $2\cdot 21$
$11$ | $2\cdot 11$ | $2\cdot 6$ | $2\cdot 20$
$12$ | $2\cdot 10$ | $2\cdot 4$ | $2\cdot 19$
$13$ | $2\cdot 8$ | $2\cdot 2$ | $2\cdot 18$
$14$ | $2\cdot 8$ | $2\cdot 2$ | $2\cdot 17$
$15$ | $2\cdot 6$ | $2\cdot 2$ | $2\cdot 16$
Table 2:  Block length $t=31$, $n=m=2$. $d_{sr}$ | dimension | Table V, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 55$ | $2\cdot 50$ | $2\cdot 59$
$5$ | $2\cdot 47$ | $2\cdot 47$ | $2\cdot 58$
$6$ | $2\cdot 46$ | $2\cdot 45$ | $2\cdot 57$
$7$ | $2\cdot 41$ | $2\cdot 40$ | $2\cdot 56$
$8$ | $2\cdot 40$ | $2\cdot 35$ | $2\cdot 55$
$9$ | $2\cdot 32$ | $2\cdot 32$ | $2\cdot 54$
$10$ | $2\cdot 32$ | $2\cdot 30$ | $2\cdot 53$
$11$ | $2\cdot 31$ | $2\cdot 27$ | $2\cdot 52$
$12$ | $2\cdot 30$ | $2\cdot 25$ | $2\cdot 51$
$13$ | $2\cdot 22$ | $2\cdot 22$ | $2\cdot 50$
$14$ | $2\cdot 22$ | $2\cdot 20$ | $2\cdot 49$
$15$ | $2\cdot 21$ | $2\cdot 15$ | $2\cdot 48$
$18$ | $2\cdot 12$ | $2\cdot 7$ | $2\cdot 45$
$22$ | $2\cdot 12$ | $2\cdot 7$ | $2\cdot 41$
$26$ | $2\cdot 7$ | $2\cdot 2$ | $2\cdot 37$
$30$ | $2\cdot 7$ | $2\cdot 2$ | $2\cdot 33$
Table 3:  Block length $t=63$, $n=m=2$. $d_{sr}$ | dimension | Table VI, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 118$ | $2\cdot 112$ | $2\cdot 123$
$5$ | $2\cdot 113$ | $2\cdot 108$ | $2\cdot 122$
$6$ | $2\cdot 112$ | $2\cdot 106$ | $2\cdot 121$
$7$ | $2\cdot 106$ | $2\cdot 100$ | $2\cdot 120$
$8$ | $2\cdot 103$ | $2\cdot 94$ | $2\cdot 119$
$9$ | $2\cdot 99$ | $2\cdot 90$ | $2\cdot 118$
$10$ | $2\cdot 98$ | $2\cdot 88$ | $2\cdot 117$
$11$ | $2\cdot 94$ | $2\cdot 82$ | $2\cdot 116$
$12$ | $2\cdot 91$ | $2\cdot 76$ | $2\cdot 115$
$13$ | $2\cdot 86$ | $2\cdot 72$ | $2\cdot 114$
$14$ | $2\cdot 85$ | $2\cdot 70$ | $2\cdot 113$
$15$ | $2\cdot 79$ | $2\cdot 64$ | $2\cdot 112$
$16$ | $2\cdot 76$ | $2\cdot 58$ | $2\cdot 111$
$17$ | $2\cdot 72$ | $2\cdot 54$ | $2\cdot 110$
$18$ | $2\cdot 72$ | $2\cdot 54$ | $2\cdot 109$
$19$ | $2\cdot 71$ | $2\cdot 54$ | $2\cdot 108$
$20$ | $2\cdot 71$ | $2\cdot 54$ | $2\cdot 107$
$21$ | $2\cdot 68$ | $2\cdot 54$ | $2\cdot 106$
$22$ | $2\cdot 67$ | $2\cdot 52$ | $2\cdot 105$
$23$ | $2\cdot 63$ | $2\cdot 50$ | $2\cdot 104$
$24$ | $2\cdot 60$ | $2\cdot 44$ | $2\cdot 103$
$25$ | $2\cdot 56$ | $2\cdot 40$ | $2\cdot 102$
$26$ | $2\cdot 56$ | $2\cdot 40$ | $2\cdot 101$
$27$ | $2\cdot 54$ | $2\cdot 38$ | $2\cdot 100$
$28$ | $2\cdot 51$ | $2\cdot 32$ | $2\cdot 99$
$29$ | $2\cdot 46$ | $2\cdot 28$ | $2\cdot 98$
$30$ | $2\cdot 46$ | $2\cdot 28$ | $2\cdot 97$
$31$ | $2\cdot 42$ | $2\cdot 26$ | $2\cdot 96$
$32$ | $2\cdot 39$ | $2\cdot 20$ | $2\cdot 95$
$38$ | $2\cdot 35$ | $2\cdot 16$ | $2\cdot 89$
$46$ | $2\cdot 29$ | $2\cdot 8$ | $2\cdot 81$
$54$ | $2\cdot 20$ | $2\cdot 2$ | $2\cdot 73$
Table 4:  Block length $t=127$, $n=m=2$. $d_{sr}$ | dimension | Table VII, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 245$ | $2\cdot 238$ | $2\cdot 251$
$5$ | $2\cdot 233$ | $2\cdot 233$ | $2\cdot 250$
$6$ | $2\cdot 232$ | $2\cdot 231$ | $2\cdot 249$
$7$ | $2\cdot 225$ | $2\cdot 224$ | $2\cdot 248$
$8$ | $2\cdot 224$ | $2\cdot 217$ | $2\cdot 247$
$9$ | $2\cdot 212$ | $2\cdot 212$ | $2\cdot 246$
$10$ | $2\cdot 211$ | $2\cdot 210$ | $2\cdot 245$
$11$ | $2\cdot 204$ | $2\cdot 203$ | $2\cdot 244$
$12$ | $2\cdot 203$ | $2\cdot 196$ | $2\cdot 243$
$13$ | $2\cdot 191$ | $2\cdot 191$ | $2\cdot 242$
$14$ | $2\cdot 190$ | $2\cdot 189$ | $2\cdot 241$
$15$ | $2\cdot 183$ | $2\cdot 182$ | $2\cdot 240$
$16$ | $2\cdot 182$ | $2\cdot 175$ | $2\cdot 239$
$19$ | $2\cdot 169$ | $2\cdot 163$ | $2\cdot 236$
$20$ | $2\cdot 168$ | $2\cdot 161$ | $2\cdot 235$
$22$ | $2\cdot 155$ | $2\cdot 154$ | $2\cdot 233$
$27$ | $2\cdot 134$ | $2\cdot 126$ | $2\cdot 228$
$28$ | $2\cdot 133$ | $2\cdot 119$ | $2\cdot 227$
$30$ | $2\cdot 120$ | $2\cdot 112$ | $2\cdot 225$
$38$ | $2\cdot 100$ | $2\cdot 91$ | $2\cdot 217$
$40$ | $2\cdot 99$ | $2\cdot 84$ | $2\cdot 215$
$46$ | $2\cdot 79$ | $2\cdot 70$ | $2\cdot 209$
$54$ | $2\cdot 65$ | $2\cdot 44$ | $2\cdot 201$
$60$ | $2\cdot 50$ | $2\cdot 35$ | $2\cdot 195$
$62$ | $2\cdot 44$ | $2\cdot 28$ | $2\cdot 193$
$76$ | $2\cdot 30$ | $2\cdot 9$ | $2\cdot 179$
$80$ | $2\cdot 30$ | $2\cdot 9$ | $2\cdot 175$
$92$ | $2\cdot 23$ | $2\cdot 9$ | $2\cdot 163$
$108$ | $2\cdot 16$ | $2\cdot 2$ | $2\cdot 147$
$120$ | $2\cdot 9$ | $2\cdot 2$ | $2\cdot 135$
$124$ | $2\cdot 9$ | $2\cdot 2$ | $2\cdot 131$
$126$ | $2\cdot 9$ | $2\cdot 2$ | $2\cdot 129$

In the case of block length $t=255$, there is no concrete sum-rank-metric code
given in [32]. We calculate the parameters of these sum-rank BCH codes
according to Theorem 9 in [32], and compare with our codes. The Singleton-like
bounds of corresponding sum-rank-metric codes are also calculated and listed
for comparison.

Table 5:  Block length $t=255$, $n=m=2$. $d_{sr}$ | dimension | Theorem 9, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 500$ | $2\cdot 492$ | $2\cdot 507$
$5$ | $2\cdot 493$ | $2\cdot 486$ | $2\cdot 506$
$6$ | $2\cdot 492$ | $2\cdot 484$ | $2\cdot 505$
$7$ | $2\cdot 484$ | $2\cdot 476$ | $2\cdot 504$
$8$ | $2\cdot 480$ | $2\cdot 468$ | $2\cdot 503$
$9$ | $2\cdot 474$ | $2\cdot 462$ | $2\cdot 502$
$10$ | $2\cdot 473$ | $2\cdot 460$ | $2\cdot 501$
$11$ | $2\cdot 468$ | $2\cdot 452$ | $2\cdot 500$
$12$ | $2\cdot 464$ | $2\cdot 444$ | $2\cdot 499$
$15$ | $2\cdot 448$ | $2\cdot 428$ | $2\cdot 496$
$16$ | $2\cdot 444$ | $2\cdot 420$ | $2\cdot 495$
$17$ | $2\cdot 438$ | $2\cdot 414$ | $2\cdot 494$
$18$ | $2\cdot 437$ | $2\cdot 412$ | $2\cdot 493$
$30$ | $2\cdot 386$ | $2\cdot 344$ | $2\cdot 481$
$31$ | $2\cdot 378$ | $2\cdot 336$ | $2\cdot 480$
$32$ | $2\cdot 374$ | $2\cdot 328$ | $2\cdot 479$
$33$ | $2\cdot 369$ | $2\cdot 322$ | $2\cdot 478$
$34$ | $2\cdot 369$ | $2\cdot 322$ | $2\cdot 477$
$35$ | $2\cdot 366$ | $2\cdot 320$ | $2\cdot 476$
$36$ | $2\cdot 364$ | $2\cdot 316$ | $2\cdot 475$
$60$ | $2\cdot 268$ | $2\cdot 192$ | $2\cdot 451$
$64$ | $2\cdot 248$ | $2\cdot 168$ | $2\cdot 447$
$68$ | $2\cdot 244$ | $2\cdot 162$ | $2\cdot 443$
$72$ | $2\cdot 239$ | $2\cdot 162$ | $2\cdot 439$
$100$ | $2\cdot 176$ | $2\cdot 112$ | $2\cdot 411$
$119$ | $2\cdot 131$ | $2\cdot 68$ | $2\cdot 392$
$120$ | $2\cdot 129$ | $2\cdot 66$ | $2\cdot 391$
$128$ | $2\cdot 103$ | $2\cdot 38$ | $2\cdot 383$
$136$ | $2\cdot 99$ | $2\cdot 32$ | $2\cdot 375$
$144$ | $2\cdot 97$ | $2\cdot 32$ | $2\cdot 367$
$199$ | $2\cdot 57$ | $2\cdot 2$ | $2\cdot 312$
$200$ | $2\cdot 57$ | $2\cdot 2$ | $2\cdot 311$
$238$ | $2\cdot 36$ | $2\cdot 2$ | $2\cdot 273$
$239$ | $2\cdot 34$ | $2\cdot 2$ | $2\cdot 272$
$240$ | $2\cdot 34$ | $2\cdot 2$ | $2\cdot 271$

Remark 2.1. As proved in Theorem 2.2, all above quaternary linear codes ${\bf
C}_{1}$ and ${\bf C}_{2}$ can exchange their positions, then we get two binary
linear sum-rank-metric codes with the same lower bounds on their dimensions
and minimum sum-rank distances. It is interesting to understand if these two
binary linear sum-rank-metric codes are equivalent.

Remark 2.2. It is well-known that for many small parameters, quaternary BCH
codes are not the presently known best codes, see [19]. Therefore larger
binary linear sum-rank-metric codes were constructed in [14].

Some lower bounds on the minimum sum-rank distances of sum-ranl BCH codes were
improved in a recent paper [1]. However our linear sum-rank-metric codes from
BCH codes over ${\bf F}_{4}$ are better than sum-rank BCH codes in [32, 1], in
particular, when the minimum sum-rank distances are larger. We speculate that
perhaps there are some unknown deep algebraic structures in these sum-rank BCH
codes of [32]. For example, are automorphism groups of these sum-rank BCH
codes of large minimum sum-rank distances larger?

In some cases, the minimum sum-rank distance $d_{sr}$ of $SR({\bf C}_{1},{\bf
C}_{2})$ is much larger than our lower bound $\min\\{d_{1},2d_{2}\\}$, as
showed in the following example.

Example 2.5. Let $\bf{C}_{1}$ be a $[25,1,25]$ narrow-sense BCH code over
$\bf{F}_{4}$ with designed distance $\delta_{1}=15$. Let $\bf{C}_{2}$ be a
$[25,2,20]$ BCH code over $\bf{F}_{4}$ with designed distance $\delta_{2}=10$
and defining set $\\{0\\},\\{1,4,6,9,11,14,16,19,\\\
21,24\\},\\{2,3,7,8,12,13,17,18,22,23\\},\\{5,20\\}$. Let $SR({\bf C}_{1},{\bf
C}_{2})$ be a block length $25$ and matrix size $2\times 2$ binary linear sum-
rank-metric code. From SageMath calculation, the minimum sum-rank distance is
actually $d_{sr}=30$.

It is well-known BCH codes are asymptotically bad. Then it is impossible to
get the asymptotically good sequence of sum-rank-metric codes from BCH codes.
Asymptotically good sequence of binary linear sum-rank-metric codes will be
constructed from Goppa codes in the next section.

###  2.2 Goppa type sum-rank-metric codes

The class of Goppa codes invented by V. D. Goppa in his 1970-1971 papers [17,
18], is one of most interesting classes of error-correcting codes and is the
origin of algebraic-geometric codes. It is well-known, that there are
sequences of binary Goppa codes meeting the Gilbert-Varshamov bound

$R(\delta)\geq 1-H_{2}(\delta),$

$H_{2}(\delta)$ is the entropy function, see [27, pp. 308-309], [27, Chapter
12] and [25, Chapter 9]. There is no the sum-rank version of Goppa codes in
the Hamming metric, as linearized Reed-Solomon codes and sum-rank BCH codes in
[28, 32]. Then it is interesting to calculate parameters of binary linear sum-
rank-metric codes of the matrix size $2\times 2$ in Theorem 2.1 from binary or
quaternary Goppa codes.

Let $G(z)$ be a polynomial in ${\bf F}_{q^{m}}[z]$ and
$L=\\{\alpha_{1},\ldots,\alpha_{n}\\}\subset{\bf F}_{q^{m}}$ be a subset of
${\bf F}_{q^{m}}$ satisfying $g(\alpha_{i})\neq 0$ for $i=1,\ldots,n$. The
Goppa code $\Gamma(L,G)$ is the set of codewords $(c_{1},\ldots,c_{n})\in{\bf
F}_{q}^{n}$ satisfying $\Sigma_{i=1}^{n}\frac{c_{i}}{z-\alpha_{i}}\equiv 0$
$mod$ $G(z)$, see [17, 18], [27, Chapter 12] and [25, Chapter 9]. This is a
subfield subcode of the GRS code with the dimension

$k(\Gamma(L,G)))\geq n-m\deg(G),$

and the minimum distance

$d(\Gamma(L,G))\geq\deg(G)+1.$

Therefore the length is flexible.

When $q=2$, this is the binary Goppa code. A binary Goppa code is more
interesting since a better lower bound on the minimum distance can be
obtained. If the Goppa polynomial $G(z)\in{\bf F}_{2^{m}}[z]$ has no multiple
roots, this is called a separable Goppa code. The minimum distance of a
separable binary Goppa code satisfies

$d(\Gamma(L,G))\geq 2\deg(G)+1,$

see [27, Theorem 6, Section3].

The typical example of binary separable Goppa codes is the irreducible Goppa
code with an irreducible Goppa polynomial $G(z)\in{\bf F}_{2^{m}}[z]$ of the
degree $r$. Then a family of binary Goppa $[2^{m},k\geq 2^{m}-rm,d\geq
2r+1]_{2}$ codes $\Gamma_{r,m}(L,G)$ was constructed, see page 345 of [27].

Theorem 2.3. Sequences of matrix size $2\times 2$ binary linear sum-rank-
metric codes of the rate

$R_{sr}(\delta_{sr})\geq
1-\frac{1}{2}(H_{4}(\delta_{sr})+H_{4}(2\delta_{sr})),$

can be constructed from binary Goppa codes, for any $\delta_{sr}$ satisfying
$0<\delta_{sr}<\frac{1}{4}$.

Proof. We take two sequences of quaternary $[t_{i},k_{i},d_{i}]_{2}$ and
$[t_{i},k_{i}^{\prime},d_{i}^{\prime}]_{2}$ Goppa codes meeting the Gilbert-
Varshamov bound, satisfying $2d_{i}=d_{i}^{\prime}$, see [25, Theorem 9.4.1].
Then

$\lim\frac{k_{i}}{t_{i}}\geq 1-H_{4}(\lim\frac{d_{i}}{t_{i}}),$

and

$\lim\frac{k_{i}^{\prime}}{t_{i}}\geq
1-H_{4}(\lim\frac{d_{i}^{\prime}}{t_{i}}).$

From Theorem 2.1, we get a sequence of linear sum-rank-metric codes of the
matrix size $2\times 2$, the sum-rank rate
$R_{sr}=\lim\frac{2(k_{i}+k_{i}^{\prime})}{4t_{i}}=\frac{1}{2}(\lim\frac{k_{i}}{t_{i}}+\lim\frac{k_{i}^{\prime}}{t_{i}})$.
The conclusion follows immediately.

In the following table we list some binary linear sum-rank-metric codes of the
block length $32$, $64$ and $128$, the matrix size $2\times 2$, from applying
two binary irreducible Goppa codes in Theorem 2.1. These codes constructed
from Theorem 2.1 applying to Hamming metric Goopa codes are called Goppa type
sum-rank-metric codes. Our codes can be comparable with corresponding sum-rank
BCH codes in [32] of the block size $31$,$63$ and $127$.

Table 6:  Block length $t$, $n=m=2$. $t$ | $d_{sr}$ | dimension | Tables, [32] | Singleton-like
---|---|---|---|---
$32$ | $5$ | $2\cdot 49$ |  $t=31$, $\dim=2\cdot 47$ | $2\cdot 60$
$32$ | $18$ | $2\cdot 12$ |  $t=31$, $\dim=2\cdot 7$ | $2\cdot 47$
$32$ | $22$ | $2\cdot 7$ |  $t=31$, $\dim=2\cdot 7$ | $2\cdot 43$
$32$ | $26$ | $2\cdot 2$ |  $t=31$, $\dim=2\cdot 2$ | $2\cdot 39$
$64$ | $5$ | $2\cdot 110$ |  $t=63$, $\dim=2\cdot 108$ | $2\cdot 124$
$128$ | $5$ | $2\cdot 235$ |  $t=127$, $\dim=2\cdot 233$ | $2\cdot 252$

It is natural to use two quaternary Goppa codes in Theorem 2.1. Some of them
have worse lower bounds on dimensions, compared with corresponding sum-rank
BCH codes in [32] of the same minimum sum-rank distances. Goppa type sum-rank-
metric codes in the following example can be comparable with sum-rank BCH
codes in [32].

Example 2.6. Let ${\bf C}_{2}$ be the $[64,37,10]_{4}$ Goppa code over ${\bf
F}_{4}$. Another code ${\bf C}_{1}$ is the $[64,52,5]_{4}$ Goppa code. From
Theorem 2.1, a binary linear sum-rank-metric code of the block length $64$,
the matrix size $2\times 2$, the minimum sum-rank distance at least $10$, and
the dimension $2\cdot(37+52)=2\cdot 89$, is constructed. This code can be
comparable with the sum-rank BCH code of the block length $63$, the dimension
$2\cdot 88$, and the same minimum sum-rank distance $10$ in [32, Table VI].
Similarly, a binary linear sum-rank-metric code of the block length $64$, the
matrix size $2\times 2$, the minimum sum-rank distance at least $14$, and the
dimension $2\cdot(25+46)=2\cdot 71$, is constructed. This code can be
comparable with the sum-rank BCH code of the block length $63$, dimension
$2\cdot 70$ and the same minimum sum-rank distance $14$ in [32, Table VI].

##  3 Linear sum-rank-metric codes from additive quaternary codes

Let us recall the following definition of additive quaternary codes in [7, 5,
4], which are the natural extension of linear quaternary codes.

Definition 3.1. Let ${\bf C}\subset{\bf F}_{4}^{n}$ be a general code. Suppose
that ${\bf C}$ satisfies the additive closure property, ${\bf c}_{1}+{\bf
c}_{2}\subset{\bf C}$, if ${\bf c}_{i}\in{\bf C}$ for $i=1,2$. We call this
code ${\bf C}$ an additive quaternary code.

A little more generalization of Theorem 2.1 is the following result. It is
obvious that we only need the “additivity” in the two Hamming metric codes
${\bf C}_{1}$ and ${\bf C}_{2}$ to preserve the “additivity” of the sum-rank-
metric code $SR({\bf C}_{1},{\bf C}_{2})$.

Theorem 3.1. Let ${\bf C}_{1}\subset{\bf F}_{4}^{t}$ and ${\bf
C}_{2}\subset{\bf F}_{4}^{t}$ be two additive $(t,4^{k_{1}},d_{1})_{4}$ and
$(t,4^{k_{2}},d_{2})_{4}$ codes. Then a block length $t$ and matrix size
$2\times 2$ binary linear sum-rank-metric code $SR({\bf C}_{1},{\bf C}_{2})$
of the minimum sum-rank distance at least $\min\\{d_{1},2d_{2}\\}$ can be
constructed explicitly. The dimension of $SR({\bf C}_{1},{\bf C}_{2})$ over
${\bf F}_{2}$ is $\dim_{{\bf F}_{2}}(SR({\bf C}_{1},{\bf
C}_{2}))=2(k_{1}+k_{2})$.

It should be noticed that the “dimension” $k$ of an additive quaternary code
may be a positive integer or the half of a positive integer, see [5].
Comparing with the presently known best quaternary linear codes in [19], some
larger additive quaternary codes were constructed in [7, 5, 4, 6]. Then the
binary linear sum-rank-metric codes constructed in [14] can be improved as the
following examples.

Example 3.1. We want to construct a binary linear sum-rank-metric code of the
block length $12$, the matrix size $2\times 2$, and the minimum sum-rank
distance $8$. If we use Theorem 2.1 or 2.2, two linear $[12,k_{1},4]_{4}$ and
$[12,k_{2},8]_{4}$ codes are needed. From [19], the largest such linear
quaternary codes are $[12,8,4]_{4}$ and $[12,3,8]_{4}$ codes. The dimension of
binary linear sum-rank-metric code constructed in Theorem 2.1 or 2.2 is of the
dimension $2\cdot(3+8)=22$. However from Table 1 of [4], the
$(12,4^{3.5},8)_{4}$ additive quaternary code was conctructed. We can
construct a binary linear sum-rank-metric code of the dimension
$2\cdot(3.5+8)=23$, with the same minimum sum-rank distance $8$.

Similarly, if we want to construct a binary linear sum-rank-metric code of the
block length $13$, the matrix size $2\times 2$, and the minimum sum-rank
distance $10$ from Theorem 2.1 or 2.2, and the presently known largest codes
in [19]. The dimension is $2\cdot(7+2)=18$. However from Table 1 of [4], the
$(13,4^{7.5},5)_{4}$ additive quaternary code was constructed. We can
construct a binary linear sum-rank-metric code of the dimension
$2\cdot(7.5+2)=19$, with the same minimum sum-rank distance $10$.

Example 3.2. Additive $(19,4^{14.5},4)_{4}$,
$(20,4^{15.5},4)_{4}$,$(21,4^{16.5},4)_{4}$ and $(22,4^{17.5},4)_{4}$ codes
were constructed in [6]. However the largest dimensions of linear quaternary
codes of length $19+i$, $i=0,1,2,3$, and the minimum distance $4$, are
$14,15,16,17$, see [19]. The largest dimensions of linear quaternary codes of
length $19,20,21,22$, and the minimum distance $8$, are $9,10,10,11$, see
[19]. Therefore the binary linear sum-rank-metric codes of the matrix size
$2\times 2$ and the block length $19,20,21,22$ constructed in Theorem 2.1 or
2.2 have their dimensions $2\cdot(9+14)=46$, $2\cdot(15+10)=50$,
$2\cdot(16+10)=52$ and $2\cdot(17+11)=56$.

From Theorem 3.1 and additive quaternary codes in [6], we construct binary
linear sum-rank-metric codes of the block length $19,20,21,22$, the matrix
size $2\times 2$, the minimum sum-rank distance $10$, and the dimensions
$47,51,53,57$.

More BCH type binary linear sum-rank-metric codes and binary linear sum-rank-
metric codes from additive quaternary codes are listed in the webpage [26] for
the convenience of readers.

##  4 Fast decoding of binary linear sum-rank-metric codes of the matrix size
$2\times 2$

In this section, we first give a reduction of the decoding in the sum-rank-
metric to several decodings in the Hamming metric. Then our reduction method
is applied to BCH and Goppa type linear sum-rank-metric codes.

###  4.1 Reduction of the decoding in the sum-rank-metric to the decoding in
the Hamming metric

The key point of the reduction of the decoding in the sum-rank-metric to the
decoding in the Hamming metric is the identification

${\bf
F}_{q}^{(n,n)}=\\{a_{0}x+a_{1}x^{q}+\cdots+a_{n-1}x^{q^{n-1}}:a_{0},\ldots,a_{n-1}\in{\bf
F}_{q^{n}}\\}.$

This is the isomorphism of the linear space over ${\bf F}_{q}$. In the case
$q=2,m=n=2$, it is of the following form,

${\bf F}_{2}^{(2,2)}=\\{a_{0}x+a_{1}x^{2}:a_{0},a_{1}\in{\bf F}_{4}\\}.$

In this paper our decoding algorithms only work for these binary linear sum-
rank-metric codes of the matrix size $2\times 2$ from one quaternary code
${\bf C}_{2}$ and one quaternary code ${\bf C}_{1}$. We give our reduction
method to reduce the decoding of the constructed binary linear sum-rank-metric
codes to the decoding of the Hamming metric codes ${\bf C}_{1}$ and ${\bf
C}_{2}$.

Let ${\bf v}=(v_{1},\ldots,v_{t})\in{\bf F}_{4}^{t}$ be a Hamming weight $t$
vector, that is, $v_{i}\neq 0$, $i=1,\ldots,t$, and ${\bf C}\subset{\bf
F}_{4}^{t}$ be a linear quaternary code, then ${\bf v}\cdot{\bf C}\subset{\bf
F}_{4}^{t}$ is the linear quaternary code defined by

${\bf v}\cdot{\bf
C}=\\{(v_{1}c_{1},\ldots,v_{t}c_{t}):(c_{1},\ldots,c_{t})\in{\bf C}\\}.$

We always represent ${\bf F}_{4}=\\{0,1,\omega,\omega^{2}\\}$, where
$\omega^{2}+\omega+1=0$. Set ${\bf 1}=(1,\ldots,1)\in{\bf F}_{4}^{t}$,
${\bf\omega}=(\omega,\ldots,\omega)\in{\bf F}_{4}^{t}$, and
${\bf\omega}^{2}=(\omega^{2},\ldots,\omega^{2})\in{\bf F}_{4}^{t}$.

For any received vector ${\bf y}\in{\bf F}_{2}^{(2,2),\ldots,(2,2)}$, of the
form

${\bf y}={\bf c}+{\bf e},$

where ${\bf c}\in SR({\bf C}_{1},{\bf C}_{2})$, and

$wt_{sr}({\bf e})\leq\left\lfloor\frac{d_{sr}-1}{4}\right\rfloor,$

it is clear that the decoding of the sum-rank-metric codes constructed in
Theorem 2.1 from two Hamming metric error-correcting codes ${\bf C}_{1}$ and
${\bf C}_{2}$ satisfying $d_{1}\geq d_{sr}$ and $d_{2}\geq\frac{d_{sr}}{2}$,
can be reduced to the decoding of these two codes ${\bf C}_{1}$ and ${\bf
C}_{2}$. However if we want to decode sum-rank-metric codes up to

$wt_{sr}({\bf e})\leq\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor,$

errors, our reduction method requires that the two quaternary linear
$[t,k_{1},d_{1}]_{4}$ code ${\bf C}_{1}$ and the $[t,k_{2},d_{2}]_{4}$ code
${\bf C}_{2}$ satisfy the condition, $d_{1}=d_{sr}$ and
$d_{2}\geq\frac{2d_{sr}}{3}$.

For any received vector ${\bf y}\in{\bf F}_{2}^{(2,2),\ldots,(2,2)}$, of the
form

${\bf y}={\bf c}+{\bf e},$

where ${\bf c}\in SR({\bf C}_{1},{\bf C}_{2})$, and

$wt_{sr}({\bf e})\leq\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor,$

we represent each coordinate matrix of ${\bf y}$ at the $i$-th position
$y_{i}=y_{0,i}x+y_{1,i}x^{2}=c_{2,i}x+c_{1,i}x^{2}+e_{0,i}x+e_{1,i}x^{2}\in{\bf
F}_{2}^{(2,2)}$. That is

${\bf y}={\bf y}_{0}x+{\bf y}_{1}x^{2}=({\bf c}_{2}+{\bf e}_{0})x+({\bf
c}_{1}+{\bf e}_{1})x^{2},$

where ${\bf c}_{1}=(c_{1,1},\ldots,c_{1,t})\in{\bf C}_{1}$, ${\bf
c}_{2}=(c_{2,1},\ldots,c_{2,t})\in{\bf C}_{2}$, ${\bf e}_{0},{\bf
e}_{1}\in{\bf F}_{4}^{t}$. Let us observe the sum-rank weight of the error
vector ${\bf e}={\bf e}_{0}x+{\bf e}_{1}x^{2}$, then

$wt_{sr}({\bf e})=2i_{1}+2i_{2}+i_{3},$

where $i_{1}$ is the number of the coordinate positions in the subset
$I_{1}\subset\\{1,\ldots,t\\}$ such that $e_{0}\neq 0$ and $e_{1}=0$, $i_{2}$
is the number of the coordinate positions in the subset
$I_{2}\subset\\{1,\ldots,t\\}$ such that $e_{0}=0$ and $e_{1}\neq 0$, and
$i_{3}$ is the number of the coordinate positions in the subset
$I_{3}\subset\\{1,\ldots,t\\}$ such that $e_{0}\neq 0$ and $e_{1}\neq 0$. It
is obvious that the Hamming weights of ${\bf e}_{0}$ and ${\bf e}_{1}$ satisfy
$wt_{H}({\bf e}_{0})=i_{1}+i_{3}\leq wt_{sr}({\bf e})$ and $wt_{H}({\bf
e}_{1})=i_{2}+i_{3}\leq wt_{sr}({\bf e})$.

First of all we decode the received word ${\bf y}_{1}\in{\bf F}_{4}^{t}$.
Since ${\bf y}_{1}={\bf c}_{1}+{\bf e}_{1}$ and

$wt_{H}({\bf
e}_{1})\leq\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor=\left\lfloor\frac{d_{1}-1}{2}\right\rfloor,$

then any decoding algorithm correcting up to half distance of ${\bf C}_{1}$
can decode ${\bf y}_{1}$ to the form ${\bf y}_{1}={\bf c}_{1}+{\bf e}_{1}$.
Therefore ${\bf e}_{1}$ is determined from the decoding algorithm of ${\bf
C}_{1}$.

We get another “received word” ${\bf y}^{\prime}={\bf y}+{\bf
c}_{1}x^{2}=({\bf c}_{2}+{\bf e}_{0})x+{\bf e}_{2}x^{2}$. Though we do not
know what the nonzero element $\frac{e_{0,i}}{e_{1,i}}$ in ${\bf F}_{4}$ is
for the coordinate position $i\in I_{3}$. There are three possibilities
$1,\omega,\omega^{2}$ for this nonzero element for each position $i\in I_{3}$.
By applying the “received word” ${\bf y}^{\prime}$ to three vectors ${\bf 1}$,
${\bf\omega}$ and ${\bf\omega}^{2}$, we get

${\bf c}_{2}+({\bf e}_{0}+{\bf e}_{1}),$ ${\bf\omega}\cdot{\bf c}_{2}+({\bf
e}_{0}\cdot{\bf\omega}+{\bf e}_{1}\cdot{\bf\omega}^{2}),$

and

${\bf\omega}^{2}\cdot{\bf c}_{2}+({\bf e}_{0}\cdot{\bf\omega}^{2}+{\bf
e}_{1}\cdot{\bf\omega}).$

Therefore for $i\in I_{3}$,

$e_{0,i}x+e_{1,i}x^{2}$

is zero for at least $\frac{i_{3}}{3}$ coordinate positions, when $x$ takes
one of three possibilities $x=1,\omega,\omega^{2}$. Then the Hamming weight of
one of three vectors

$({\bf e}_{0}+{\bf e}_{1}),$ $({\bf e}_{0}\cdot{\bf\omega}+{\bf
e}_{1}\cdot{\bf\omega}^{2}),$

and

$({\bf e}_{0}\cdot{\bf\omega}^{2}+{\bf e}_{1}\cdot{\bf\omega}).$

is at most

$i_{1}+i_{2}+\frac{2i_{3}}{3}=\frac{wt_{sr}({\bf
e})}{2}+\frac{i_{3}}{6}\leq\frac{2wt_{sr}({\bf
e})}{3}\leq\left\lfloor\frac{d_{2}-1}{2}\right\rfloor,$

from the assumption $d_{2}\geq\frac{2d_{sr}}{3}$.

By applying any decoding algorithm of the code ${\bf C}_{2}$, correcting up to
$\left\lfloor\frac{d_{2}-1}{2}\right\rfloor$ errors, to three equivalent codes
${\bf C}_{2}$, ${\bf\omega}\cdot{\bf C}_{2}$ and ${\bf\omega}^{2}\cdot{\bf
C}_{2}$, on three “received words”

${\bf y}^{\prime}(1)={\bf y}({\bf 1})+{\bf c}_{1},$ ${\bf
y}^{\prime}({\bf\omega})={\bf y}({\bf\omega})+{\bf\omega}^{2}\cdot{\bf
c}_{1},$

and

${\bf y}^{\prime}({\bf\omega}^{2})={\bf
y}({\bf\omega}^{2})+{\bf\omega}\cdot{\bf c}_{1},$

we can decode to determine the whole error ${\bf e}$.

Theorem 4.1. Let $SR({\bf C}_{1},{\bf C}_{2})$ be a block length $t$ and
matrix size $2\times 2$ binary linear sum-rank-metric code of the minimum sum-
rank distance $d_{sr}$, constructed from two quaternary $[t,k_{i},d_{i}]_{4}$
codes ${\bf C}_{i}$, $i=1,2$ satisfying $d_{1}\geq d_{sr}$ and
$d_{2}\geq\frac{2d_{sr}}{3}$ as in Theorem 2.1. Suppose that there are
decoding algorithms correcting up to
$\left\lfloor\frac{d_{i}-1}{2}\right\rfloor$ errors for the code ${\bf C}_{i}$
with the cost ${\bf T}_{i}$, $i=1,2$. Then by applying the decoding algorithm
to the code ${\bf C}_{1}$ once, and the the decoding algorithm to three
equivalent codes ${\bf C}_{2}$, ${\bf\omega}\cdot{\bf C}_{2}$ and
${\bf\omega}^{2}\cdot{\bf C}_{2}$ three times, we can decode the sum-rank-
metric code $SR({\bf C}_{1},{\bf C}_{2})$ up to
$\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors. The cost of this
decoding algorithm in the sum-rank-metric is

${\bf T}_{1}+3{\bf T}_{2}.$

It is obvious that binary linear sum-rank-metric codes constructed in Theorem
2.2 can be decoded similarly, if the condition $d_{1}\geq\frac{2d_{sr}}{3}$ is
assumed. The following examples show that our reduction of the decoding in the
sum-rank-metric to the decoding in the Hamming metric works well.

Example 4.1. Let $\bf{C}_{1}$ be a $[31,10,12]$ BCH code over $\bf{F}_{4}$
with designed distance $\delta_{1}=12$ and defining set
$\\{0\\},\\{1,2,4,8,16\\},\\{3,6,12,17,24\\},\\{5,9,10\\\
,18,20\\},\\{7,14,19,25,28\\}$. Let $\bf{C}_{2}$ be a $[31,15,8]$ BCH code
over $\bf{F}_{4}$ with designed distance $\delta_{2}=8$ and defining set
$\\{0\\},\\{1,2,4,8,16\\},\\{3,6,12,17,24\\},\\{5\\\ ,9,10,18,20\\}$. Let
$SR({\bf C}_{1},{\bf C}_{2})$ be a block length $31$ and matrix size $2\times
2$ binary linear sum-rank-metric code of the minimum sum-rank distance
$d_{sr}\geq 12$. Take
$\mathbf{c}_{1}=(0,w+1,w,1,w,w,w+1,1,w,1,w+1,0,1,1,0,w+1,0,0,w+1,0,w,w+1,0,1,w,w,w,w+1,w+1,1,1)\in\mathbf{C}_{1}$,
$\mathbf{c}_{2}=(w,w+1,w,1,0,0,w,w+1,w,0,w,0,w+1,w,w,1,1,w+1,0,w+1,w,0,1,w,0,0,w,\\\
w+1,1,1,0)\in\mathbf{C}_{2}$,
$\mathbf{e}_{1}=(0,0,0,w+1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\\\
w,0,0,0,0,0,0,0,1)$, and
$\mathbf{e}_{2}=(0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,\\\
0,0,1,0,0,0,0,0,0,0,w)$. Then $wt_{H}(\mathbf{e}_{1})=3$,
$wt_{H}(\mathbf{e}_{2})=4$, and
$wt_{st}(\mathbf{e})=5\leq\lfloor{\frac{d_{sr}-1}{2}}\rfloor$. Using the
decoding command in SageMath, we can get $\mathbf{c}_{1}$ from
$\mathbf{y}=\mathbf{c}_{1}+\mathbf{e}_{1}$. Since
$wt_{H}(\mathbf{e}_{2})=4>\lfloor\frac{8-1}{2}\rfloor=3$, we can not get
$\mathbf{c}_{2}$ from a direct decoding $\mathbf{c_{2}}+\mathbf{e}_{2}$. By
decoding three words $\mathbf{y}^{\prime}(1)$, $\mathbf{y}^{\prime}(w)$ and
$\mathbf{y}^{\prime}(w^{2})$, we get $\mathbf{c}_{2}$ from
$\mathbf{y}^{\prime}(w)$ with the Hamming weight
$wt_{H}(\omega\mathbf{e}_{2}+\omega^{2}\mathbf{e}_{1})=2$ of the error vector.
Then we decode the above received word of $SR({\bf C}_{1},{\bf C}_{2})$.

Example 4.2. Let $\bf{C}_{1}$ be a $[63,30,15]$ narrow-sense BCH code over
$\bf{F}_{4}$ with designed distance $\delta_{1}=15$. Let $\bf{C}_{2}$ be a
$[63,42,10]$ narrow-sense BCH code over $\bf{F}_{4}$ with designed distance
$\delta_{2}=10$. Let $SR({\bf C}_{1},{\bf C}_{2})$ be a block length $63$ and
matrix size $2\times 2$ binary linear sum-rank-metric code of the minimum sum-
rank distance $d_{sr}\geq 15$. Take $\mathbf{c}_{1}=(0,w,1,w,w,1,0,w,1,0,w\\\
,w,0,w,0,w,w,w,1,0,1,w,1,1,w+1,w,w+1,w+1,w+1,0,0,w+1,w+1,w,1,1,w,1,1,1,0,0,w+1,0,1,w,w,w,0,w,w,0,0,1,1,1,w+1,w,1,w+1,w+1,w,w+1)\in\mathbf{C}_{1}$,
$\mathbf{c}_{2}=(1,w+1,0,1,0,1,1,w+1,1,w,w+1,0,0,1,1,w+1,1,w,1,w,1,w+1,w,0,w+1,w+1,0,w+1,0,1,w,1,1,w+1,0,1,w,0,0,0,1\\\
,w+1,1,w+1,w,0,w,w,w,0,w+1,w,w+1,w,0,w+1,1,w,w,0,0,w+1,0)\in\mathbf{C}_{2}$,
$\mathbf{e}_{1}=(0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,\\\
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\\\ ,0)$,
and $\mathbf{e}_{2}=(0,w+1,0,0,0,0,w,0,0,0,0,0,0,0,0,0,0,0,0,0,w,0,0,0,0,0,\\\
0,0,0,0,0,0,0,0,w,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0\\\
,0,0)$. Then $wt_{H}(\mathbf{e}_{1})=3$, $wt_{H}(\mathbf{e}_{2})=5$, and
$wt_{st}(\mathbf{e})=7\leq\lfloor{\frac{d_{sr}-1}{2}}\rfloor$. Using the
decoding command in SageMath, we can get $\mathbf{c}_{1}$ from
$\mathbf{y}=\mathbf{c}_{1}+\mathbf{e}_{1}$. Since
$wt_{H}(\mathbf{e}_{2})=5>\lfloor\frac{10-1}{2}\rfloor=4$, we can not get
$\mathbf{c}_{2}$ from the decoding $\mathbf{c_{2}}+\mathbf{e}_{2}$. By
decoding three words $\mathbf{y}^{\prime}(1)$, $\mathbf{y}^{\prime}(w)$ and
$\mathbf{y}^{\prime}(w^{2})$, we get $\mathbf{c}_{2}$ from
$\mathbf{y}^{\prime}(w)$ with the Hamming weight
$wt_{H}(\omega\mathbf{e}_{2}+\omega^{2}\mathbf{e}_{1})=3$ of the error vector.
Hence, we decode the above received word of $SR({\bf C}_{1},{\bf C}_{2})$.

Example 4.3. Let $\bf{C}_{1}$ be a $[63,22,24]$ BCH code over $\bf{F}_{4}$
with designed distance $\delta_{1}=24$ and defining set
$\\{0\\},\\{1,4,16\\},\\{2,8,32\\},\\{3,12,48\\},\\{5,17,\\\
20\\},\\{6,24,33\\},\\{7,28,49\\},\\{9,18,36\\},\\{10,34,40\\},\\{11,44,50\\},\\{13,19,52\\},\\\
\\{14,35,56\\},\\{15,51,60\\},\\{21\\},\\{22,25,37\\}$. Let $\bf{C}_{2}$ be a
$[63,29,16]$ BCH code over $\bf{F}_{4}$ with designed distance $\delta_{2}=16$
and defining set $\\{0\\},\\{1,4,16\\},\\\
\\{2,8,32\\},\\{3,12,48\\},\\{5,17,20\\},\\{6,24,33\\},\\{7,28,49\\},\\{9,18,36\\},\\{10,34,40\\}\\\
,\\{11,44,50\\},\\{13,19,52\\},\\{14,35,56\\}$. Let $SR({\bf C}_{1},{\bf
C}_{2})$ be a block length $63$ and matrix size $2\times 2$ binary linear sum-
rank-metric code of the minimum sum-rank distance $d_{sr}\geq 24$. Take
$\mathbf{c}_{1}=(0,w,w+1,w,w+1,w+1,w,1,0,w,w,0,w,0,w,w,w,0,0,w,0,1,w+1,w,1,1,0,w+1,w,1,w,w+1,0,w+1,w,w,w+1,1,1,w,1,w,1,0,w,1,w+1,0,w,1,w,1,w+1,w,0,w+1,1,0,w,1,1,w,0)\in\mathbf{C}_{1}$,
$\mathbf{c}_{2}=(1,1,0,0,w+1,0,1,w,w+1,w,w,w+1,w,w+1,0,1,w,w,1,w,w,1,0,w,w,0,0,w+1,1,0,w,1,1,w+1,w,w+1,w+1,0,1,w,w+1,w+1,w,w+1,0,w+1,w,1,w,w,0,1,w+1,0,0,w+1,1,0,1,0,w+1,w+1,w)\in\mathbf{C}_{2}$,
$\mathbf{e}_{1}=(1,0,w+1,0,w,0,1,0,w+1,0,w,0,1,0\\\
,w+1,0,w,0,1,0,w+1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\\\
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, and
$\mathbf{e}_{2}=(1,0,1,0,1,0,1,0,1,0,1,0,\\\
1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\\\
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$. Then $wt_{H}(\mathbf{e}_{1})=11$,
$wt_{H}(\mathbf{e}_{2})=11$, and
$wt_{st}(\mathbf{e})=11\leq\lfloor{\frac{d_{sr}-1}{2}}\rfloor$. Using the
decoding command in SageMath, we can get $\mathbf{c}_{1}$ from
$\mathbf{y}=\mathbf{c}_{1}+\mathbf{e}_{1}$. Since
$wt_{H}(\mathbf{e}_{2})=11>\lfloor\frac{16-1}{2}\rfloor=7$, we can not get
$\mathbf{c}_{2}$ from the decoding $\mathbf{c_{2}}+\mathbf{e}_{2}$. We can
decode three words $\mathbf{y}^{\prime}(1)$, $\mathbf{y}^{\prime}(w)$ and
$\mathbf{y}^{\prime}(w^{2})$, and get $\mathbf{c}_{2}$ from
$\mathbf{y}^{\prime}(1)$ with the Hamming weight
$wt_{H}(\mathbf{e}_{2}+\mathbf{e}_{1})=7$ of the error vector. Hence, we
decode the above received word of $SR({\bf C}_{1},{\bf C}_{2})$.

###  4.2 Fast decoding algorithms for BCH type binary linear sum-rank-metric
codes

The following binary linear sum-rank-metric codes of the block length $15,63$,
$255$, and the matrix size $2\times 2$, are constructed from two quaternary
BCH $[t,k_{i},d_{i}]_{4}$ codes, satisfying $d_{1}=d_{sr}$ and
$d_{2}\geq\frac{2d_{sr}}{3}$. From Theorem 2.1, the constructed binary linear
sum-rank-metric codes have their minimum sum-rank distances at least
$\min\\{d_{1},2d_{2}\\}=d_{sr}$. Though these codes are smaller than BCH type
sum-rank-metric codes in Section 2, by comparing to these sum-rank BCH codes
in [32], our codes are larger when the minimum sum-rank distances are the
same. Therefore it is good to give fast decoding algorithms of these BCH type
binary linear sum-rank-metric codes.

Table 7:  Block length $t=15$, $n=m=2$. $d_{sr}$ | dimension | Table IV, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 22$ | $2\cdot 20$ | $2\cdot 27$
$5$ | $2\cdot 19$ | $2\cdot 18$ | $2\cdot 26$
$6$ | $2\cdot 18$ | $2\cdot 16$ | $2\cdot 25$
$7$ | $2\cdot 16$ | $2\cdot 14$ | $2\cdot 24$
$8$ | $2\cdot 13$ | $2\cdot 10$ | $2\cdot 23$
$9$ | $2\cdot 12$ | $2\cdot 8$ | $2\cdot 22$
$10$ | $2\cdot 11$ | $2\cdot 8$ | $2\cdot 21$
$11$ | $2\cdot 8$ | $2\cdot 6$ | $2\cdot 20$
$12$ | $2\cdot 7$ | $2\cdot 4$ | $2\cdot 19$
$13$ | $2\cdot 5$ | $2\cdot 2$ | $2\cdot 18$
$14$ | $2\cdot 5$ | $2\cdot 2$ | $2\cdot 17$
$15$ | $2\cdot 5$ | $2\cdot 2$ | $2\cdot 16$
Table 8:  Block length $t=63$, $n=m=2$. $d_{sr}$ | dimension | Table VI, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 115$ | $2\cdot 112$ | $2\cdot 123$
$5$ | $2\cdot 110$ | $2\cdot 108$ | $2\cdot 122$
$6$ | $2\cdot 109$ | $2\cdot 106$ | $2\cdot 121$
$7$ | $2\cdot 104$ | $2\cdot 100$ | $2\cdot 120$
$8$ | $2\cdot 100$ | $2\cdot 94$ | $2\cdot 119$
$9$ | $2\cdot 98$ | $2\cdot 90$ | $2\cdot 118$
$10$ | $2\cdot 94$ | $2\cdot 88$ | $2\cdot 117$
$11$ | $2\cdot 88$ | $2\cdot 82$ | $2\cdot 116$
$12$ | $2\cdot 85$ | $2\cdot 76$ | $2\cdot 115$
$13$ | $2\cdot 81$ | $2\cdot 72$ | $2\cdot 114$
$14$ | $2\cdot 79$ | $2\cdot 70$ | $2\cdot 113$
$20$ | $2\cdot 62$ | $2\cdot 54$ | $2\cdot 107$
$22$ | $2\cdot 58$ | $2\cdot 52$ | $2\cdot 105$
$24$ | $2\cdot 51$ | $2\cdot 44$ | $2\cdot 103$
$28$ | $2\cdot 43$ | $2\cdot 32$ | $2\cdot 99$
$30$ | $2\cdot 41$ | $2\cdot 28$ | $2\cdot 97$
$38$ | $2\cdot 28$ | $2\cdot 16$ | $2\cdot 89$
$46$ | $2\cdot 17$ | $2\cdot 8$ | $2\cdot 81$
$54$ | $2\cdot 9$ | $2\cdot 2$ | $2\cdot 73$
$60$ | $2\cdot 9$ | $2\cdot 2$ | $2\cdot 67$
Table 9:  Block length $t=255$, $n=m=2$. $d_{sr}$ | dimension | Theorem 9, [32] | Singleton-like
---|---|---|---
$4$ | $2\cdot 496$ | $2\cdot 492$ | $2\cdot 507$
$5$ | $2\cdot 489$ | $2\cdot 486$ | $2\cdot 506$
$6$ | $2\cdot 488$ | $2\cdot 484$ | $2\cdot 505$
$7$ | $2\cdot 481$ | $2\cdot 476$ | $2\cdot 504$
$8$ | $2\cdot 476$ | $2\cdot 468$ | $2\cdot 503$
$9$ | $2\cdot 473$ | $2\cdot 462$ | $2\cdot 502$
$10$ | $2\cdot 468$ | $2\cdot 460$ | $2\cdot 501$
$11$ | $2\cdot 460$ | $2\cdot 452$ | $2\cdot 500$
$12$ | $2\cdot 456$ | $2\cdot 444$ | $2\cdot 499$
$15$ | $2\cdot 444$ | $2\cdot 428$ | $2\cdot 496$
$16$ | $2\cdot 436$ | $2\cdot 420$ | $2\cdot 495$
$17$ | $2\cdot 429$ | $2\cdot 414$ | $2\cdot 494$
$18$ | $2\cdot 428$ | $2\cdot 412$ | $2\cdot 493$
$30$ | $2\cdot 372$ | $2\cdot 344$ | $2\cdot 481$
$31$ | $2\cdot 365$ | $2\cdot 336$ | $2\cdot 480$
$32$ | $2\cdot 360$ | $2\cdot 328$ | $2\cdot 479$
$33$ | $2\cdot 358$ | $2\cdot 322$ | $2\cdot 478$
$34$ | $2\cdot 354$ | $2\cdot 322$ | $2\cdot 477$
$35$ | $2\cdot 348$ | $2\cdot 320$ | $2\cdot 476$
$36$ | $2\cdot 346$ | $2\cdot 316$ | $2\cdot 475$
$60$ | $2\cdot 242$ | $2\cdot 192$ | $2\cdot 451$
$64$ | $2\cdot 222$ | $2\cdot 168$ | $2\cdot 447$
$68$ | $2\cdot 212$ | $2\cdot 162$ | $2\cdot 443$
$72$ | $2\cdot 203$ | $2\cdot 162$ | $2\cdot 439$
$100$ | $2\cdot 138$ | $2\cdot 112$ | $2\cdot 411$
$119$ | $2\cdot 116$ | $2\cdot 68$ | $2\cdot 392$
$120$ | $2\cdot 114$ | $2\cdot 66$ | $2\cdot 391$
$128$ | $2\cdot 99$ | $2\cdot 38$ | $2\cdot 383$
$136$ | $2\cdot 88$ | $2\cdot 32$ | $2\cdot 375$
$144$ | $2\cdot 75$ | $2\cdot 32$ | $2\cdot 367$
$199$ | $2\cdot 18$ | $2\cdot 2$ | $2\cdot 312$
$200$ | $2\cdot 18$ | $2\cdot 2$ | $2\cdot 311$
$238$ | $2\cdot 17$ | $2\cdot 2$ | $2\cdot 273$
$239$ | $2\cdot 17$ | $2\cdot 2$ | $2\cdot 272$
$240$ | $2\cdot 17$ | $2\cdot 2$ | $2\cdot 271$

In general, the binary linear sum-rank-metric codes of the block length
$4^{h}-1$, the matrix size $2\times 2$, constructed from two BCH quaternary
codes ${\bf C}_{1}$ and ${\bf C}_{2}$ satisfying $d_{1}\geq d_{sr}$ and
$d_{2}\geq\frac{d_{sr}}{3}$ as in Theorem 2.1, are larger than sum-rank BCH
codes in [32] of the same minimum sum-rank distances. For these better BCH
type binary linear sum-rank-metric codes, we have the following result from
Theorem 4.1.

Corollary 4.1. Let $SR({\bf C}_{1},{\bf C}_{2})$ be a block length $t$ and
matrix size $2\times 2$ BCH type binary linear sum-rank-metric code of the
minimum sum-rank distance $d_{sr}$, constructed from two quaternary BCH
$[t,k_{i},d_{i}]_{4}$ codes ${\bf C}_{i}$, $i=1,2$ as in Theorem 2.1,
satisfying $d_{1}\geq d_{sr}$ and $d_{2}\geq\frac{2d_{sr}}{3}$. Then by
applying any BCH decoding algorithm to the BCH code ${\bf C}_{1}$ once and the
three equivalent BCH codes ${\bf C}_{2}$, ${\bf\omega}\cdot{\bf C}_{2}$ and
${\bf\omega}^{2}\cdot{\bf C}_{2}$ three times, we can decode the sum-rank-
metric code $SR({\bf C}_{1},{\bf C}_{2})$ up to
$\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors.

Therefore for all above BCH type linear sum-rank-metric codes better than sum-
rank BCH codes invented in [32], we give the decoding algorithm, which can
correct up to $\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors. The
complexity is the same as the Welch-Berlemkamp algorithm given in [28, 32].
Since our decoding in the sum-rank-metric is reduced to the decoding in the
Hamming metric, it seems that our decoding can be implemented practically by
the using of fast BCH decoders in the Hamming metric.

###  4.3 Fast decoding algorithms for Goppa type binary linear sum-rank-metric
codes

By applying Theorem 4.1 and the Sugiyama decoding algorithm, see [45] and [23,
Section 5.4.3], to the binary linear sum-rank-metric codes of the matrix size
$2\times 2$, constructed from two Goppa binary or quaternary codes satisfying
$d_{1}=d_{sr}$ and $d_{2}\geq\frac{2d_{sr}}{3}$ as in Theorem 2.1, we get the
fast decoding algorithm of Goppa type binary linear sum-rank-metric codes of
the matrix size $2\times 2$. These Goppa type binary linear sum-rank-metric
codes can be decoded up to $\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$
errors, within the complexity $O(t^{2})$ operations in ${\bf F}_{4}$.

On the other hand, a little weaker version of asymptotically good binary
linear sum-rank-metric codes of the matrix size $2\times 2$ in Theorem 2.2 can
be fastly decoded to $\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors,
within the complexity $O(t^{2})$ operations in ${\bf F}_{4}$.

Theorem 4.2. Sequences of matrix size $2\times 2$ binary linear sum-rank-
metric codes of the rate

$R_{sr}(\delta_{sr})\geq
1-\frac{1}{2}(H_{4}(\frac{4}{3}\delta_{sr})+H_{4}(2\delta_{sr})),$

can be constructed from binary Goppa codes, for any $\delta_{sr}$ satisfying
$0<\delta_{sr}<\frac{1}{4}$. These binary linear sum-rank-metric codes can be
decoded up to $\left\lfloor\frac{d_{sr}-1}{2}\right\rfloor$ errors, within the
complexity $O(t_{i}^{2})$ operations in ${\bf F}_{4}$, where $t_{i}$ is the
block length of the sum-rank-metric code.

###  4.4 Decoders in the sum-rank-metric from fast decoders in the Hamming
metric

If we use two length $31$ quaternary BCH $[31,k_{i},d_{i}]_{4}$ codes,
$i=1,2$, satisfying $d_{1}\geq d_{sr}$ and $d_{2}\geq\frac{2d_{sr}}{3}$ in
Theorem 2.1 to construct binary linear sum-rank-metric codes of the block
length $31$, the dimension is not bigger than that of the corresponding sum-
rank BCH codes. For example, when $d_{sr}=12$, we get a dimension $2\times 25$
binary linear sum-rank-metric code from the $[31,15,8]_{4}$ BCH code and the
$[31,10,12]_{4}$ BCH code. The sum-rank BCH code of the same minimum sum-rank
distance $d_{sr}=12$ in Table V of [32] has the same dimension $2\times 25$.
From Theorem 4.1, our fast decoder for our code has the same complexity as the
Welch-Berlemkamp decoder in [32]. However by combining various fast decoders
in the Hamming metric, actually we can decode larger binary linear sum-rank-
metric code of the same minimum sum-rank distance with the same complexity.

By padding zero to the extended QR (quadratic residue) quaternary
$[30,15,12]_{4}$ code, we get a quaternary $[31,15,12]_{4}$ code. Then by
applying Theorem 2.1 to this code and the quaternary $[31,15,8]_{4}$ BCH code,
a binary linear sum-rank-metric code of the block length $31$, the matrix size
$2\times 2$ and the minimum sum-rank distance $12$ is constructed. The
dimension of this code over ${\bf F}_{2}$ is $2\cdot(15+15)=2\cdot 30$. This
code is much larger than the corresponding dimension $2\cdot 25$ sum-rank BCH
code of the same minimum sum-rank distance $12$ in Table V of [32].

From Theorem 4.1, by combining the Welch-Berlemkamp decoder for the extended
QR code and the BCH quaternary code, the fast decoder for above our
constructed binary linear sum-rank-metric code of the dimension $2\cdot 30$
can be obtained. This decoding algorithm has the same complexity as the Welch-
Berlekamp decoder in [32]. In [14], we showed that binary linear sum-rank-
metric codes of the matrix size $2\times 2$ from presently best known codes in
[19] are much larger than sum-rank BCH codes of the same minimum sum-rank
distances. If the second Hamming metric quaternary codes in Tables 1-10 of
[14] are replaced by a quaternary code satisfying
$d_{2}\geq\frac{2d_{sr}}{3}$, much larger binary linear sum-rank-metric codes
than sum-rank BCH codes of the same minimum sum-rank distances can still be
constructed explicitly. By applying our reduction result Theorem 4.1, it leads
to fast decoding algorithms for these better binary linear sum-rank-metric
codes of the matrix size $2\times 2$, if fast decoders for these presently
known best codes in [19] are available. As far as we know, there are fast
decoding algorithms for many presently known best Hamming metric codes in
[19].

This example illustrates that Theorem 4.1 about reducing the decoding in the
sum-rank-metric to the decoding in the Hamming metric can be applied to better
sum-rank-metric codes constructed from various Hamming metric codes. When
there are fast decoding algorithms for these Hamming metric codes, then it
leads to the fast decoding algorithm to the larger sum-rank-metric codes
directly. Moreover, considering the the implementation using the FFT (fast
Fourier transformation), it seems that our decoding algorithms in the sum-
rank-metric is more practical than these decoding algorithms developed in [28,
32, 2]. It should be remarked that Theorem 3.1 is only for binary linear sum-
rank-metric codes of the matrix size $2\times 2$.

##  5 Conclusion

In this paper, the new construction and fast decoding of binary linear sum-
rank-metric codes from the combination of Hamming metric codes and
$q$-polynomials were proposed. Many BCH and Goppa type binary linear sum-rank-
metric codes have larger dimensions when compared with previous sum-rank BCH
codes of the same minimum sum-rank distances. We also improved our
construction of binary linear sum-rank-metric codes of the matrix size
$2\times 2$ from additive quaternary codes. We showed that the decoding of BCH
and Goppa type binary linear sum-rank-metric codes can be reduced to the
decoding of BCH and Goppa quaternary codes in the Hamming metric. Then fast
decoding algorithms of these BCH and Goppa type sum-rank-metric codes with
better parameters were presented. Asymptotically good sequences of quadratic-
time encodable and decodable binary linear sum-rank-metric codes of the matrix
size $2\times 2$ were also constructed. The reduction of decoding in the sum-
rank-metric to the decoding in the Hamming metric is general and can be
applied to any Hamming metric error-correcting codes with fast decodings, such
as algebraic geometric codes, see [14, Section II.D]. It is an interesting
open problem that if the decoding of the sum-rank-metric codes from Theorem
2.1, can be reduced to the decodings of ${\bf C}_{1}$ and ${\bf C}_{2}$,
without the assumption $d_{2}\geq\frac{2d_{sr}}{3}$.

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