Asymptotically good binary linear codes with asymptotically good self-intersection spans

TL;DR

This paper constructs a family of binary linear codes that are asymptotically good and whose self-intersection spans are also asymptotically good, using algebraic geometry, concatenation, and bilinear algebra techniques.

Contribution

It introduces a novel construction of asymptotically good binary codes with good self-intersection spans, leveraging algebraic geometry and field algebra innovations.

Findings

Constructed asymptotically good binary codes with good self-intersection spans.

Used algebraic-geometry codes and concatenation techniques.

Applied bilinear algebra and field extension properties.

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.