Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids

TL;DR

This paper explores the relationship between the algebraic invariants of Stanley-Reisner rings and the combinatorial properties of matroids derived from linear codes, focusing on Hamming weights and Betti numbers.

Contribution

It establishes how generalized Hamming weights of codes can be inferred from Betti numbers of associated Stanley-Reisner rings of matroids.

Findings

Betti numbers encode information about Hamming weights

Matroid invariants relate to algebraic properties of Stanley-Reisner rings

New methods to determine code parameters from algebraic invariants

Abstract

To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid.