The square @math of a linear error correcting code @math is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in @math . Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both @math and @math . In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes @math with a relatively large dimension of @math and minimum distance of the square @math are discussed. In some cases, the constructions lead to codes @math such that both @math and @math simultaneously have the largest possible minimum distances for their length and dimensions.
However, as mentioned above, Reed-Solomon codes have the restriction that @math . Therefore the asymptotic behaviour of families of squares of codes has been considered, where the finite field @math is fixed and @math grows to infinity. The existence, over every finite field, of asymptotically good families We say that a family of codes @math with lengths @math is asymptotically good if @math when @math and the limits @math and @math exist and are strictly positive. of codes whose squares also form an asymptotically good family was established in @cite_0 . For small fields, this result requires a combination of an algebraic geometric construction over a sufficiently large (but constant) extension field and a special concatenation function to achieve a final construction over the small finite field. However, @cite_26 showed that families of codes with such asymptotic properties are not very abundant, since choosing codes uniformly at random (among all codes of a prescribed dimension that grows linearly with the length) will, with high probability, not satisfy the desired properties.
@cite_0 @cite_26
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If C is a binary linear code, let C〈2〉 be the linear code spanned by intersections of pairs of codewords of C. We construct an asymptotically good family of binary linear codes such that, for C ranging in this family, C〈2〉 also form an asymptotically good family. For this, we use algebraic-geometry codes, concatenation, and a fair amount of bilinear algebra. More precisely, the two main ingredients used in our construction are, first, a description of the symmetric square of an odd degree extension field in terms only of field operations of small degree, and second, a recent result of Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an odd degree extension field. Given a linear code @math , one can define the @math -th power of @math as the span of all componentwise products of @math elements of @math . A power of @math may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code "typically" fill the whole space? We give a positive answer, for codes of dimension @math and length roughly @math or smaller. Moreover, the convergence speed is exponential if the difference @math is at least linear in @math . The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros.