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 d1, · · · , d k of a matroid M are determined by the 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 N-graded Betti numbers of the Stanley-Reisner ring. The negative examples all come from matroids of type M (C).
Generalizing polynomials previously studied in the context of linear codes,
we define weight polynomials and an enumerator for a matroid $M$. Our main
result is that these polynomials are determined by Betti numbers associated
with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and
so-called elongations of $M$. Generalizing Greene's theorem from coding theory,
we show that the enumerator of a matroid is equivalent to its Tutte polynomial.Comment: 21 page
We define generalized Hamming weights for almost affine codes. We show that this definition is natural since we can extend some well known properties of the generalized Hamming weights for linear codes, to almost affine codes. In addition we discuss duality of almost affine codes, and of the smaller class of multilinear codes.
Given a constant weight linear code, we investigate its weight hierarchy and the Stanley-Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weight.
We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of Fq. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C.
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