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

Alessandro Neri, Paolo Santonastaso, and Ferdinando Zullo

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Submission date: 11. Feb. 2022
Bibtex
MSC-Numbers: 11T71, 51E20, 11T06, 94B05
Keywords and phrases: Sum-rank metric codes, one-weight codes, linear sets, simplex codes, linearized Reed-Solomon codes
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We provide a geometric characterization of k-dimensional ?_q^m linear sum-rank metric codes as tuples of ?_q-subspaces of (?_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 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.