  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.
