Asymptotically Good Binary Linear Codes With Asymptotically Good Self-Intersection Spans

Abstract

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, the 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.

On Squares of Cyclic Codes

“…Therefore the asymptotic behaviour of families of squares of codes has been considered, where the finite field F q is fixed and n grows to infinity. The existence, over every finite field, of asymptotically good families 1 of codes whose squares also form an asymptotically good family was established in [26]. 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.…”

“…Instead of considering the asymptotic setting, this paper focuses on specific values for the length of the code C; here the problem is that not many existing results that can be applied to, for example, the setting of linear binary codes with lengths, say, n ≤ 10000. One option is to use Reed-Solomon codes over large enough extension fields paired with the concatenation technique in [26]. Reed-Muller codes are a family of binary codes for which it is relatively easy to determine the minimum distance of their squares.…”

Squares of Random Linear Codes

“…Motivated in part by these applications, asymptotically good codes whose squares are also asymptotically good (and we impose no conditions on the duals) have been shown to exist for all finite fields in [30]. This construction carefully combines algebraic geometric codes that have asymptotically good higher powers, which can be constructed over large enough finite fields, with a field descent concatenation technique.…”

Powers of codes and applications to cryptography

“…However, it is not known if one can extend this result to the remaining finite fields, as the concatenation in [3] does not preserve this property. Nevertheless, if we do not worry about the duals but only about the squares, then there is the following result [30].…”