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.