Decoding up to Hartmann-Tzeng and Roos bounds for rank codes

TL;DR

This paper introduces a new class of linear block codes that generalize existing codes, providing bounds on their rank distance, decoding algorithms, and conditions for decoding up to these bounds, including subfield subcodes and interleaved codes.

Contribution

It defines a new class of codes generalizing Gabidulin and skew cyclic codes, with new bounds, decoding algorithms, and conditions, plus analysis of subfield subcodes and interleaved codes.

Findings

Established Hartmann-Tzeng-like and Roos-like bounds for the new codes.

Developed decoding algorithms up to the new bounds.

Showed equivalence of subfield subcodes and interleaved codes with respect to the rank metric.

Abstract

A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann-Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.