# American Institute of Mathematical Sciences

May  2017, 11(2): 267-282. doi: 10.3934/amc.2017018

## Rank equivalent and rank degenerate skew cyclic codes

 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220, Denmark

Received  February 2016 Revised  March 2016 Published  May 2017

Fund Project: The author is supported by The Danish Council for Independent Research (Grant No. DFF-4002-00367).

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. 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.

Citation: Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
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