On Representing Matroids via Modular Independence

TL;DR

This paper explores a matrix-based approach to representing matroids over local rings using modular independence, establishing conditions for when these systems are matroids and analyzing their properties and limitations.

Contribution

It introduces a new framework for matroid representation over local rings, providing criteria for matroidhood, duality, and bounds, including representations over rings not realizable over fields.

Findings

Chain rings are characterized by monotone minimal generators.

Puncturing corresponds to deletion in codes over finite chain rings.

The uniform matroid $U_{2, n}$ is representable iff n is within a specific bound.

Abstract

We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a matroid. Under a mild nilpotent hypothesis, we show that chain rings are exactly the local rings for which the minimal number of generators is monotone on finitely generated submodules, and over commutative chain rings we obtain a criterion for the associated independence system to be a matroid. For codes over finite commutative chain rings, we identify puncturing with deletion, show that shortening agrees with contraction under a contractibility hypothesis, and establish duality for free codes. We further derive bounds for simple and uniform matroids, prove that the uniform matroid $U_{2, n}$ is representable if and only if the size $n$ is at most the sum of the cardinalities of the local ring and its unique maximal ideal, and show that all excluded minors for $\mathbb{F}_{4}$-representability are representable over $\mathbb{Z}/4\mathbb{Z}$. The examples also include ring representations of matroids not representable over any field, such as the V\'{a}mos matroid over $\mathbb{Z}/8\mathbb{Z}$.