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Kernels and ranks of cyclic and negacyclic quaternary codes

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Abstract

We study the rank and kernel of \({\mathbb {Z}}_4\) cyclic codes of odd length n and give bounds on the size of the kernel and the rank. Given that a cyclic code of odd length is of the form \(\mathcal { C}= \langle fh, 2fg \rangle \), where \(fgh=x^n-1\), we show that \(\langle 2f \rangle \subseteq \mathcal { K}(\mathcal { C}) \subseteq \mathcal { C}\) and \(\mathcal { C}\subseteq \mathcal { R}(\mathcal { C}) \subseteq \langle fh, 2g \rangle \) where \(\mathcal { K}(\mathcal { C}) \) is the preimage of the binary kernel and \(\mathcal { R}(\mathcal { C})\) is the preimage of the space generated by the image of \(\mathcal { C}\). Additionally, we show that both \(\mathcal { K}(\mathcal { C})\) and \(\mathcal { R}(\mathcal { C})\) are cyclic codes and determine \(\mathcal { K}(\mathcal { C})\) and \(\mathcal { R}(\mathcal { C})\) in numerous cases. We conclude by using these results to determine the case for negacyclic codes as well.

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References

  1. Blackford T.: Cyclic codes over \({\mathbb{Z}}_4\) of oddly even length. International Workshop on Coding and Cryptography (WCC 2001) (Paris). Discret. Appl. Math. 128(1), 27–46 (2003).

  2. Blackford T.: Negacyclic codes over \({\mathbb{Z}}_4\) of even length. IEEE Trans. Inf. Theory 49(6), 1417–1424 (2003).

  3. Blackford T.: Negacyclic duadic codes. Finite Fields Appl. 14(4), 930–943 (2008).

  4. Borges J., Phelps K.P., Rifà J., Zinoviev V.: On \({\mathbb{Z}}_4\)-linear preparata-like and Kerdock-like. IEEE Tans. Inf. Theory 49(11), 2834–2843 (2003).

  5. Calderbank A.R., Sloane N.J.A.: Modular and \(p\)-adic cyclic codes. Des. Codes Cryptogr. 6(1), 21–35 (1995).

  6. Dougherty S.T., Ling S.: Cyclic codes over \({\mathbb{Z}}_4\) of even length. Des. Codes Cryptogr. 39(2), 127–153 (2006).

  7. 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).

  8. Fernández-Córdoba C., Pujol J., Villanueva M.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-linear codes: rank and kernel. Des. Codes Cryptogr. 56(1), 43–59 (2010).

  9. 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(2), 301–319 (1994).

  10. Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over \({\mathbb{Z}}_4\). IEEE Trans. Inf. Theory 42(5), 1594–1600 (1996).

  11. Pless V.S., Solé P., Qian Z.: Cyclic self-dual \({\mathbb{Z}}_4\)-codes. Finite Field Appl. 3(1), 48–69 (1997).

  12. Wolfmann J.: Binary images of cyclic codes over \({\mathbb{Z}}_4\). IEEE Tans. Inf. Theory 47(5), 1773–1779 (2001).

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Acknowledgments

This work has been partially supported by the Spanish MICINN Grant TIN2013-40524-P and by the Catalan AGAUR Grant 2014SGR-691.

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Correspondence to Steven T. Dougherty.

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Communicated by A. Winterhof.

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Dougherty, S.T., Fernández-Córdoba, C. Kernels and ranks of cyclic and negacyclic quaternary codes. Des. Codes Cryptogr. 81, 347–364 (2016). https://doi.org/10.1007/s10623-015-0163-6

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  • DOI: https://doi.org/10.1007/s10623-015-0163-6

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