Rank equivalent and rank degenerate skew cyclic codes

TL;DR

This paper investigates rank equivalences between skew cyclic codes of different lengths, introduces new length concepts, and characterizes rank degenerate codes, extending known results to all lengths and skew cyclic codes.

Contribution

It introduces new length measures for skew cyclic codes, characterizes rank degenerate codes, and shows the minimal length rank equivalent code can be a pseudo-skew cyclic code.

Findings

Rank equivalences between skew cyclic codes of different lengths are characterized.

New length concepts for skew cyclic codes are defined and computed.

The minimal length rank equivalent code can be a pseudo-skew cyclic code.

Abstract

Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric, where lengths of codes correspond to the number of outgoing links from the source when applying the code on a network. We study rank equivalences between skew cyclic codes of different lengths and, with the aim of finding the skew cyclic code of smallest length that is rank equivalent to a given one, we define different types of length for a given skew cyclic code, relate them and compute them in most cases. We give different characterizations of rank degenerate skew cyclic codes using conventional polynomials and linearized polynomials. Some known results on the rank weight hierarchy of cyclic codes for some lengths are obtained as particular cases and extended to all lengths and to all skew cyclic codes. Finally, we prove that the smallest length of a linear code that is rank equivalent to a given skew cyclic code can be attained by a pseudo-skew cyclic code. Throughout the paper, we find new relations between linear skew cyclic codes and their Galois closures.