On the roots and minimum rank distance of skew cyclic codes

TL;DR

This paper explores the structure and bounds of skew cyclic codes in rank metric, establishing their root space descriptions, lattice relations, and extending classical bounds to the rank-metric context.

Contribution

It introduces root space and lattice descriptions of skew cyclic codes, extends classical bounds to rank-metric codes, and links these codes to Hamming-metric cyclic codes.

Findings

Lattice of skew cyclic codes is anti-isomorphic to root space lattice

Extended rank-BCH, van Lint-Wilson's shift, and Hartmann-Tzeng bounds to rank metric

Skew cyclic codes over the base field include all Hamming cyclic codes

Abstract

Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint-Wilson's shift and Hartmann-Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include all Hamming-metric cyclic codes, giving then a new relation between these codes and rank-metric skew cyclic codes.