A generalization of weight polynomials to matroids

Abstract

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

Pure resolutions, linear codes, and Betti numbers

“…More recent work of Johnsen, Roksvold and Verdure [18] shows that the Betti numbers of C and its elongations determine the so-called generalized weight polynomial of C. Thus, if we combine this with the results of Jurrius and Pellikaan [19], then we obtain a direct relation between the generalized weight enumerator of C and the Betti numbers of C and of its elongations. It is clear therefore that explicit determination of Betti numbers of codes would be useful and interesting.…”

“…We remark that H c and c determine each other. In other words, if c, d ∈ P k−1 (F q 2 ), then: H c = H d ⇔ c = d. Now from [9, Theorem 3.1] and from Theorem 7.4 as well as Theorem 8.1 (and its corollary) of [7], we see that (18) |H c ∩ P k−2 | =    n k−1 if H c is not a tangent hyperplane,…”

A generalization of Kung’s theorem

“…In [4], one expresses the polynomials from the present Definition 1 in terms of the Betti numbers of the Stanley-Reisner rings of the matroid associated to the parity check matrix of the code in question, and so-called elongations of this matroid. The matroids are then thought of simplicial complexes with facets the bases of the matroids.…”

Linear codes over signed graphs

“…If C is an [m, k]-linear code, then δ 1 (C) < · · · < δ k (C) [18,42]. The following duality theorem of Wei is a classical result in this area.…”

“…The generalized Hamming weights (GHWs) of a linear code are parameters of interest in many applications [12,16,20,27,31,37,42,43,45] and they have been nicely related to the graded Betti numbers of the ideal of cocircuits of the matroid of a linear code [19,20], to the nullity function of the dual matroid of a linear code [42], and to the enumerative combinatorics of linear codes [3,18,22,23]. Because of this, their study has attracted considerable attention, but determining them is in general a difficult problem.…”