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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
7/2022

The geometry of one-weight codes in the sum-rank metric

Alessandro Neri, Paolo Santonastaso and Ferdinando Zullo

Abstract

We provide a geometric characterization of $k$-dimensional $\mathbb F_{q^m}$ linear sum-rank metric codes as tuples of $\mathbb F_q$-subspaces of $(\mathbb F_{q^m})^k$. We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when $k=2$, they are one-weight, as in the Hamming-metric case. We then focus on constant rank profile codes in the sum-rank metric, which are a special family of one weight codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the $n$-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of $\mathrm{GL}(k,q^m)$. They turn out to be constant rank-profile -- and hence one-weight -- and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on $2$-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when $q=2$.

Received:
Feb 11, 2022
Published:
Feb 14, 2022
MSC Codes:
11T71, 51E20, 11T06, 94B05
Keywords:
Sum-rank metric codes, one-weight codes, linear sets, simplex codes, linearized Reed-Solomon codes

Related publications

inJournal
2023 Repository Open Access
Alessandro Neri, Paolo Santonastao and Ferdinando Zullo

The geometry of one-weight codes in the sum-rank metric

In: Journal of combinatorial theory / A, 194 (2023) 1, p. 105703