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.
Similar content being viewed by others
References
Boucher D., Ulmer F.: Coding with skew polynomial rings. J. Symb. Comput. 44(12), 1644–1656 (2009) (Gröbner Bases in Cryptography, Coding Theory, and Algebraic Combinatorics).
Boucher D., Geiselmann W., Ulmer F.: Skew-cyclic codes. Appl. Algebr. Eng. Commun. Comput. 18(4), 379–389 (2007).
Chaussade L., Loidreau P., Ulmer F.: Skew codes of prescribed distance or rank. Des. Codes Cryptogr. 50(3), 267–284 (2009).
Delsarte P.: On subfield subcodes of modified reed-solomon codes (corresp.). IEEE Trans. Inf. Theory 21(5), 575–576 (2006).
Ducoat J., Oggier F.: Rank weight hierarchy of some classes of cyclic codes. In: Information Theory Workshop (ITW), 2014 IEEE, pp. 142–146 (2014).
Duursma I.M., Pellikaan R.: A symmetric Roos bound for linear codes. J. Comb. Theory, Ser. A 113(8), 1677–1688 (2006) (Special Issue in Honor of Jacobus H. van Lint).
Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21(1), 3–16 (1985).
Gabidulin, E.M.: Rank q-cyclic and pseudo-q-cyclic codes. In: IEEE International Symposium on Information Theory ISIT 2009, pp. 2799–2802 (2009).
Gursoy F., Siap I., Yildiz B.: Construction of skew cyclic codes over \({\mathbb{F}}_q+v{\mathbb{F}}_q\). Adv. Math. Commun. 8(3), 313–322 (2014).
Hartmann C.R.P., Tzeng K.K.: Generalizations of the BCH bound. Inf. Control 20(5), 489–498 (1972).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Kshevetskiy, A., Gabidulin, E.M.: The new construction of rank codes. In: Proceedings of the International Symposium on Information Theory ISIT 2005, pp. 2105–2108 (2005).
Lidl, R., Niederreiter, H.: Finite Fields, vol. 20. Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Amsterdam (1983).
Martínez-Peñas U.: On the similarities between generalized rank and Hamming weights and their applications to network coding. IEEE Trans. Inf. Theory 62(7), 4081–4095 (2016).
Ore O.: On a special class of polynomials. Trans. Am. Math. Soc. 35(3), 559–584 (1933).
Ore O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933).
Pellikaan, R.: The shift bound for cyclic, Reed-Muller and geometric Goppa codes. In: Arithmetic, Geometry and Coding Theory, vol. 4, pp. 155–174. Luminy (1996).
Silva D., Kschischang F.R.: On metrics for error correction in network coding. IEEE Trans. Inf. Theory 55(12), 5479–5490 (2009).
Sripati, U., Rajan, B.S.: On the rank distance of cyclic codes. In: Proceedings of the IEEE International Symposium on Information Theory, June (2003).
Stichtenoth H.: On the dimension of subfield subcodes. IEEE Trans. Inf. Theory 36(1), 90–93 (1990).
van Lint J., Wilson R.: On the minimum distance of cyclic codes. IEEE Trans. Inf. Theory 32(1), 23–40 (1986).
Acknowledgments
The author wishes to thank the anonymous reviewers for their very helpful comments and suggestions. The author also wishes to thank Olav Geil, Ruud Pellikaan and Diego Ruano for fruitful discussions and careful reading of the manuscript. Finally, the author gratefully acknowledges the support from The Danish Council for Independent Research (Grant No. DFF-4002-00367).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Jungnickel.
Rights and permissions
About this article
Cite this article
Martínez-Peñas, U. On the roots and minimum rank distance of skew cyclic codes. Des. Codes Cryptogr. 83, 639–660 (2017). https://doi.org/10.1007/s10623-016-0262-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-016-0262-z