It is well-known in coding theory that the minimum distance of a code is a matroidal invariant; namely the smallest size of a circuit of the linear matroid induced by the dual code. Thus the minimum distance of a code is the smallest degree of a squarefree monomial in the Stanley-Reisner ideal of the matroid of the dual code. In the 1970s the minimum distance of a code was generalized to the sequence of $r$-th \textit{generalized Hamming weights} -- in this language the minimum distance of a code is the first generalized Hamming weight. Nowadays the sequence of generalized Hamming weights is known as the \textit{weight hierarchy} of a code, and it is a highly studied object in coding theory. In this talk we first show that 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. 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. Thus the 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. In particular, we show how the Waldschmidt constant of the Stanley-Reisner ideal of a matroid can be expressed in terms of the generalized Hamming weights. Time permitting, we apply our results to projective varieties known as matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel.
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