# 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.

## References

SHOWING 1-10 OF 21 REFERENCES

### Duality of codes supported on regular lattices, with an application to enumerative combinatorics

- Mathematics, Computer ScienceDes. Codes Cryptogr.
- 2018

A general class of regular weight functions on finite abelian groups are introduced, and the theory of MacWilliams identities are applied to enumerative combinatorics problems, obtaining closed formulae for the number of rectangular matrices over a finite having prescribed rank and satisfying some linear conditions.

### Code Enumerators and Tutte Polynomials

- Computer ScienceIEEE Transactions on Information Theory
- 2010

A very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity is proved and is applied to codeword m-tuples.

### Minimal Vectors in Linear Codes

- Computer ScienceIEEE Trans. Inf. Theory
- 1998

It is shown that for even codes the set of zero neighbors is strictly optimal in this class of algorithms, which implies that general asymptotic improvements of the zero-neighbors algorithm in the frame of gradient-like approach are impossible.

### On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes

- MathematicsArXiv
- 2021

A complete characterization of the purity of graded minimal free resolutions of StanleyReisner rings associated to generalized Reed-Muller codes of an arbitrary order is given.

### The Matroid of Supports of A Linear Code

- Mathematics, Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 1997

This paper proves analogous results for the support weight distributions of a code with relation between the Hamming weight enumerator of a linear code and the Tutte polynomial of the corresponding matroid.

### Generalized Hamming Weights for Linear Codes

- Computer Science
- 2001

This work studies d3(C) and higher Hamming weights for BCH(2m, 5) codes by a close examination of the words of weight 5 and proves that the second generalized Hamming weight d2(C).

### Hamming weights and Betti numbers of Stanley–Reisner rings associated to matroids

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2012

This work shows how the weights of a matroid M are determined by the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of $$M$$, and derives some consequences.

### Theory of Error-correcting Codes

- Computer Science

This course expounds the principles of coded modulations for the Gaussian channel and, if time permits, for Rician and Rayleigh fading channels (fully interleaved), and reminds students of the basics of the theory of linear codes for conventional memoryless ergodic channels.