Abstract
A binary linear code C is a \({\mathbb {Z}}_2\)-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the \({\mathbb {Z}}_2[x]\)-module \({\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).\) We determine the structure of \({\mathbb {Z}}_2\)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of \({\mathbb {Z}}_2\)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between \({{\mathbb {Z}}}_2\)-double cyclic and other families of cyclic codes, and show some examples of distance optimal \({\mathbb {Z}}_2\)-double cyclic codes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abualrub T., Siap I., Aydin N.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes. IEEE Trans. Inf. Theory 60, 1508–1514 (2014).
Aydogdu I., Abualrub T., Siap I.: On \({\mathbb{Z}}_2{\mathbb{Z}}_2[u]\)-additive codes. Int. J. Comput. Math. 92(9), 1806–1814 (2015).
Aydogdu I., Abualrub T., Siap I.: \(\mathbb{Z}_2\mathbb{Z }_2[u]\)-cyclic and constacyclic codes. IEEE Trans. Inf. Theory. doi:10.1109/TIT.2016.2632163.
Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54, 167–179 (2010).
Borges J., Fernández-Córdoba C., Ten-Valls R.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes, generator polynomials and dual codes. IEEE Trans. Inf. Theory 62, 6348–6354 (2016).
Fernández-Córdoba C., Pujol J., Villanueva M.: On Rank and Kernel of \({\mathbb{Z}}_4\)-Linear Codes. Lecture Notes in Computer Science, vol. 5228, pp. 46–55. Springer, Berlin (2008).
Fernández-Córdoba C., Pujol J., Villanueva M.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-linear codes: rank and kernel. Des. Codes Cryptogr. 56, 43–59 (2010).
Grassl M.: Table of bounds on linear codes. [Online]. Available: http://www.codestable.de (1995).
Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z}}_4\)-linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York (1977).
Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over \({\mathbb{Z}}_4\). IEEE Trans. Inf. Theory 42, 1594–1600 (1996).
Siap I., Kulhan N.: The structure of generalized quasi cyclic codes. Appl. Math. E-Notes 5, 24–30 (2005).
Vega G., Wolfmann J.: Some families of \({\mathbb{Z}}_4\)-cyclic codes. Finite Fields Appl. 10, 530–539 (2004).
Wolfmann J.: Binary images of cyclic codes over \({\mathbb{Z}}_4\). IEEE Trans. Inf. Theory 47, 1773–1779 (2001).
Acknowledgements
This work has been partially supported by the Spanish MINECO Grants TIN2016-77918-P and MTM2015-69138-REDT, and by the Catalan AGAUR Grant 2014SGR-691.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Wolfmann.
Rights and permissions
About this article
Cite this article
Borges, J., Fernández-Córdoba, C. & Ten-Valls, R. \({{\mathbb {Z}}}_2\)-double cyclic codes. Des. Codes Cryptogr. 86, 463–479 (2018). https://doi.org/10.1007/s10623-017-0334-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-017-0334-8