# Generalized weights of codes over rings and invariants of monomial ideals

@inproceedings{Gorla2022GeneralizedWO, title={Generalized weights of codes over rings and invariants of monomial ideals}, author={Elisa Gorla and Alberto Ravagnani}, year={2022} }

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 Fq-linear codes endowed with the Hamming metric, the ideal coincides…

## 2 Citations

### Free Resolutions and Generalized Hamming Weights of binary linear codes

- Computer Science, MathematicsMathematics
- 2022

It is proved that the first and second generalized Hamming weights of a binary linear code can be computed from a set of monomials associated with a binomial ideal related with the code.

### Generalized weights of convolutional codes

- Computer ScienceArXiv
- 2022

This paper proposes a new deﬁnition of generalized weights of convolutional codes, that takes into account the underlying module structure of the code, and derives the basic properties of the authors' generalized weights.

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