Generalized weights of codes over rings and invariants of monomial ideals

TL;DR

This paper develops an algebraic framework linking generalized weights of codes over rings to the graded Betti numbers of associated monomial ideals, extending known results from finite fields to more general rings.

Contribution

It introduces a support-based algebraic theory for codes over rings and connects generalized weights to monomial ideal invariants, generalizing prior results from finite fields.

Findings

Generalized weights can be derived from Betti numbers of associated monomial ideals.

In the case of finite fields, the ideal matches the Stanley-Reisner ideal of the code's matroid.

Many codes are generated by their minimal support codewords.

Abstract

We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a code. Our main result states that, under suitable assumptions, the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal. In the case of $\mathbb{F}_q$-linear codes endowed with the Hamming metric, the ideal coincides with the Stanley-Reisner ideal of the matroid associated to the code via its parity-check matrix. In this special setting, we recover the known result that the generalized weights of an $\mathbb{F}_q$-linear code can be obtained from the graded Betti numbers of the ideal of the matroid associated to the code. We also study subcodes and codewords of minimal support in a code, proving that a large class of $R$-linear codes is generated by its codewords of minimal support.