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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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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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