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Designs, Codes and Cryptography

Erschienen in:

31.12.2016

There is exactly one \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic 1-perfect code

verfasst von: Joaquim Borges, Cristina Fernández-Córdoba

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2017

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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.

Vorheriger Artikel Mapping prefer-opposite to prefer-one de Bruijn sequences

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Titel: There is exactly one -cyclic 1-perfect code
verfasst von: Joaquim Borges
Cristina Fernández-Córdoba
Publikationsdatum: 31.12.2016
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2017
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-016-0323-3

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There is exactly one \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic 1-perfect code

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