Abstract
Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.
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Acknowledgements
The authors thank the anonymous referees for their valuable comments, which enabled them to improve the quality of the paper. 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.
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Communicated by G. McGuire.
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Borges, J., Fernández-Córdoba, C. There is exactly one \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic 1-perfect code. Des. Codes Cryptogr. 85, 557–566 (2017). https://doi.org/10.1007/s10623-016-0323-3
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DOI: https://doi.org/10.1007/s10623-016-0323-3