# American Institute of Mathematical Sciences

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
##### References:
 [1] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. [2] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008. [3] J. Ducoat, Generalized rank weights: a duality statement, in Topics in Finite Fields, (2015), 101-109. doi: 10.1090/conm/632/12622. [4] J. Ducoat and F. Oggier, Rank weight hierarchy of some classes of cyclic codes, in 2014 IEEE Inf. Theory Workshop, (2014), 142-146. [5] E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. [6] E. M. Gabidulin, Rank q-cyclic and pseudo-q-cyclic codes, in IEEE Int. Symp. Inf. Theory, (2009), 2799-2802. [7] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. [8] J. Kurihara, R. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inf. Theory, 61 (2015), 3912-3936. doi: 10.1109/TIT.2015.2429713. [9] R. Lidl and H. Niederreiter, Finite Fields Addison-Wesley, Amsterdam, 1983. [10] U. Martínez-Peñas, On the roots and minimum rank distance of skew cyclic codes, Des. Codes Crypt., 83 (2017), 639-660. doi: 10.1007/s10623-016-0262-z. [11] U. Martínez-Peñas, On the similarities between generalized rank and Hamming weights and their applications to network coding, IEEE Trans. Inf. Theory, 62 (2016), 4081-4095. doi: 10.1109/TIT.2016.2570238. [12] O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849. [13] O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508. doi: 10.2307/1968173. [14] D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inf. Theory, 55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817. [15] H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inf. Theory, 36 (1990), 90-93. doi: 10.1109/18.50376.

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##### References:
 [1] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. [2] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008. [3] J. Ducoat, Generalized rank weights: a duality statement, in Topics in Finite Fields, (2015), 101-109. doi: 10.1090/conm/632/12622. [4] J. Ducoat and F. Oggier, Rank weight hierarchy of some classes of cyclic codes, in 2014 IEEE Inf. Theory Workshop, (2014), 142-146. [5] E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. [6] E. M. Gabidulin, Rank q-cyclic and pseudo-q-cyclic codes, in IEEE Int. Symp. Inf. Theory, (2009), 2799-2802. [7] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. [8] J. Kurihara, R. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inf. Theory, 61 (2015), 3912-3936. doi: 10.1109/TIT.2015.2429713. [9] R. Lidl and H. Niederreiter, Finite Fields Addison-Wesley, Amsterdam, 1983. [10] U. Martínez-Peñas, On the roots and minimum rank distance of skew cyclic codes, Des. Codes Crypt., 83 (2017), 639-660. doi: 10.1007/s10623-016-0262-z. [11] U. Martínez-Peñas, On the similarities between generalized rank and Hamming weights and their applications to network coding, IEEE Trans. Inf. Theory, 62 (2016), 4081-4095. doi: 10.1109/TIT.2016.2570238. [12] O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849. [13] O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508. doi: 10.2307/1968173. [14] D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inf. Theory, 55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817. [15] H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inf. Theory, 36 (1990), 90-93. doi: 10.1109/18.50376.
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