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Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

Michael DiPasquale, Louiza Fouli, Arvind Kumar, Ştefan O. Tohǎneanu

Abstract

It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for $r$). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design.

Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

Abstract

It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the -th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the -th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for ). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design.
Paper Structure (23 sections, 41 theorems, 107 equations)

This paper contains 23 sections, 41 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on matroids, codes, and ideals
  3. Matroids
  4. Matroids and coding theory
  5. Simplicial complexes and ideals associated to a matroid
  6. Symbolic powers and the Waldschmidt constant
  7. Generalized Hamming weights of matroids and symbolic powers
  8. Subadditivity and duality for finite sequences
  9. The initial degree sequence
  10. Duality for increasing sequences
  11. Sequence of Generalized Hamming weights of matroids and their Wei dual
  12. Finitely Generated Rees Algebras and Matroids
  13. Noetherian Rees algebras
  14. Generators for the symbolic Rees algebra of the Stanley-Reisner ideal of a matroid
  15. Matroids with subadditive and cosubadditive generalized Hamming weights
  16. ...and 8 more sections

Key Result

Theorem 2.1

$($We$)$ Let $\mathcal{C}$ be an $[n,k,d]$-linear code with a parity check matrix $H$. For any $j\in [n]$, let $H_j$ denote the $j$-th column of $H$. Then for any $r\in [k]$,

Theorems & Definitions (108)

  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 98 more