Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

Abstract: It is well-known that the first generalized Hamming weight of a code, more commonly called \textit{the minimum distance} of the code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a code is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid of the dual code (in the appropriate range for $r$). It turns out that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. This implies that generalized Hamming weights -- which can be defined in a natural way for matroids -- are fundamentally tied to the structure of symbolic powers of Stanley-Reisner ideals of matroids. We illustrate this by studying initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of generalized Hamming weights and working out many examples that are meaningful from a coding-theoretic perspective. Our results also apply to projective varieties known as matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel.