Designs, Codes and Cryptography

, Volume 83, Issue 3, pp 639–660 | Cite as

On the roots and minimum rank distance of skew cyclic codes



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.


Cyclic codes Finite rings Hamming distance Linearized polynomial rings Rank distance Skew cyclic codes 

Mathematics Subject Classification

15A03 15B33 94B15 



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).


  1. 1.
    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).Google Scholar
  2. 2.
    Boucher D., Geiselmann W., Ulmer F.: Skew-cyclic codes. Appl. Algebr. Eng. Commun. Comput. 18(4), 379–389 (2007).Google Scholar
  3. 3.
    Chaussade L., Loidreau P., Ulmer F.: Skew codes of prescribed distance or rank. Des. Codes Cryptogr. 50(3), 267–284 (2009).Google Scholar
  4. 4.
    Delsarte P.: On subfield subcodes of modified reed-solomon codes (corresp.). IEEE Trans. Inf. Theory 21(5), 575–576 (2006).Google Scholar
  5. 5.
    Ducoat J., Oggier F.: Rank weight hierarchy of some classes of cyclic codes. In: Information Theory Workshop (ITW), 2014 IEEE, pp. 142–146 (2014).Google Scholar
  6. 6.
    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).Google Scholar
  7. 7.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21(1), 3–16 (1985).Google Scholar
  8. 8.
    Gabidulin, E.M.: Rank q-cyclic and pseudo-q-cyclic codes. In: IEEE International Symposium on Information Theory ISIT 2009, pp. 2799–2802 (2009).Google Scholar
  9. 9.
    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).Google Scholar
  10. 10.
    Hartmann C.R.P., Tzeng K.K.: Generalizations of the BCH bound. Inf. Control 20(5), 489–498 (1972).Google Scholar
  11. 11.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).Google Scholar
  12. 12.
    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).Google Scholar
  13. 13.
    Lidl, R., Niederreiter, H.: Finite Fields, vol. 20. Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Amsterdam (1983).Google Scholar
  14. 14.
    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).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ore O.: On a special class of polynomials. Trans. Am. Math. Soc. 35(3), 559–584 (1933).Google Scholar
  16. 16.
    Ore O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933).Google Scholar
  17. 17.
    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).Google Scholar
  18. 18.
    Silva D., Kschischang F.R.: On metrics for error correction in network coding. IEEE Trans. Inf. Theory 55(12), 5479–5490 (2009).Google Scholar
  19. 19.
    Sripati, U., Rajan, B.S.: On the rank distance of cyclic codes. In: Proceedings of the IEEE International Symposium on Information Theory, June (2003).Google Scholar
  20. 20.
    Stichtenoth H.: On the dimension of subfield subcodes. IEEE Trans. Inf. Theory 36(1), 90–93 (1990).Google Scholar
  21. 21.
    van Lint J., Wilson R.: On the minimum distance of cyclic codes. IEEE Trans. Inf. Theory 32(1), 23–40 (1986).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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