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

Abstract

To each linear code $C$ over a finite field we associate the matroid $M(C)$ of its parity check matrix. For any matroid $M$ one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type $M(C)$, these weights are the same as those of the code $C$. In our main result we show how the weights $d_1,\ldots ,d_k$ of a matroid $M$ are determined by the $\mathbb N$-graded Betti numbers of the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of $M$, and derive some consequences. We also give examples which give negative results concerning other types of (global) Betti numbers, and using other examples we show that the generalized Hamming weights do not in general determine the $\mathbb N$-graded Betti numbers of the Stanley–Reisner ring. The negative examples all come from matroids of type $M(C)$.