Generalized weight polynomials -- and how they are determined by Betti numbers.

Abstract:

Let C be a linear code over the field F_q. This code may be extended to a code C' over F_Q, where Q=q^m. Let A(Q)_i be the number of words in C' of weight i, and let M(C) be the vector matroid associated to a parity-check matrix of C. We use Hochster's formula to show that the Betti numbers belonging to a minimal free resolution of the Stanley-Reisner ideal associated to M(C) and its so-called elongations determine A(Q)_i.

This is joint work with Trygve Johnsen and Hugues Verdure.