Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes - Archive ouverte HAL Access content directly
Journal Articles SIAM Journal on Applied Algebra and Geometry Year : 2021

## Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes

Daniel Augot
• Function : Author
• PersonId : 833459
Alain Couvreur
• Function : Author
• PersonId : 926260
Julien Lavauzelle

#### Abstract

This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in $\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-F\"uredi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $\mathbb{L}$ is a Kummer extension.

### Dates and versions

hal-02882019 , version 1 (26-06-2020)

### Identifiers

• HAL Id : hal-02882019 , version 1
• ARXIV :
• DOI :

### Cite

Daniel Augot, Alain Couvreur, Julien Lavauzelle, Alessandro Neri. Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes. SIAM Journal on Applied Algebra and Geometry, 2021, 5 (2), pp.165-199. ⟨10.1137/20M1348583⟩. ⟨hal-02882019⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

112 View