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

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.