Sum-rank product codes and bounds on the minimum distance

TL;DR

This paper explores the tensor product of codes with Hamming and rank metrics to create sum-rank metric codes, introduces a group-theoretic framework for cyclic-skew-cyclic codes, and generalizes bounds on minimum distance.

Contribution

It introduces a new construction of sum-rank metric codes via tensor products and provides a group-theoretic characterization of cyclic-skew-cyclic codes, along with generalized bounds on minimum distance.

Findings

Established a generalization of Roos and Hartmann-Tzeng bounds for sum-rank metric.

Provided a new lower bound on the minimum distance of product codes.

Connected cyclic-skew-cyclic codes to tensor product constructions.

Abstract

The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic. A group theoretical description of these codes is given, after investigating the semilinear isometries in the sum-rank metric. Finally, a generalization of the Roos and the Hartmann-Tzeng bounds for the sum rank-metric is established, as well as a new lower bound on the minimum distance of one of the two codes constituting the product code.