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We show that the generalized minimum distance function is non-increasing as the degree varies for reduced standard graded algebras over a field. This allows us to define its regularity index and its stabilization value. The stabilization value is computed for every cases. We study how the regularity index varies as the number of polynomial increases, and use this to give bounds for it.

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.