We provide a geometric characterization of k-dimensional Fqm-linear sum-rank metric codes as tuples of Fq-subspaces of Fqkm. 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, qm). 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.(c) 2022 Elsevier Inc. All rights reserved.
The geometry of one-weight codes in the sum-rank metric
Santonastaso, P;Zullo, F
2023
Abstract
We provide a geometric characterization of k-dimensional Fqm-linear sum-rank metric codes as tuples of Fq-subspaces of Fqkm. 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, qm). 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.(c) 2022 Elsevier Inc. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
