Abstract
We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes.
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Acknowledgements
The authors thank Prof. Josep Rifà for valuable comments on automorphism groups of linear codes.
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Communicated by J. Bierbrauer.
This work has been partially supported by the Spanish MINECO Grants TIN2016-77918-P (AEI/FEDER, UE) and MTM2015-69138-REDT, and by the Catalan AGAUR Grant 2014SGR-691.
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Borges, J., Fernández-Córdoba, C. A characterization of \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u]\)-linear codes. Des. Codes Cryptogr. 86, 1377–1389 (2018). https://doi.org/10.1007/s10623-017-0401-1
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DOI: https://doi.org/10.1007/s10623-017-0401-1
Keywords
- \(\mathbb {Z}_2\)-linear codes
- \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-linear codes
- \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes