There is exactly one Z<inf>2</inf>Z<inf>4</inf> -cyclic 1-perfect code

Joaquim Borges; Cristina Fernández-Córdoba

doi:10.1007/s10623-016-0323-3

There is exactly one Z<inf>2</inf>Z<inf>4</inf> -cyclic 1-perfect code

Joaquim Borges, Cristina Fernández-Córdoba

Research output: Contribution to journal › Article › Research › peer-review

2 Citations (Scopus)

Abstract

© 2016, Springer Science+Business Media New York. Let C be a Z2Z4-additive code of length n> 3. We prove that if the binary Gray image of C is a 1-perfect nonlinear code, then C cannot be a Z2Z4-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 Z2Z4-additive 1-perfect code gives a Z2Z4-additive extended 1-perfect code. We also prove that such a code cannot be Z2Z4-cyclic.

Original language	English
Pages (from-to)	557-566
Journal	Designs, Codes, and Cryptography
Volume	85
Issue number	3
DOIs	https://doi.org/10.1007/s10623-016-0323-3
Publication status	Published - 1 Dec 2017

Keywords

Perfect codes
Simplex codes
Z Z -additive cyclic codes 2 4

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title = "There is exactly one Z2Z4 -cyclic 1-perfect code",

abstract = "{\textcopyright} 2016, Springer Science+Business Media New York. Let C be a Z2Z4-additive code of length n> 3. We prove that if the binary Gray image of C is a 1-perfect nonlinear code, then C cannot be a Z2Z4-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 Z2Z4-additive 1-perfect code gives a Z2Z4-additive extended 1-perfect code. We also prove that such a code cannot be Z2Z4-cyclic.",

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author = "Joaquim Borges and Cristina Fern{\'a}ndez-C{\'o}rdoba",

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T1 - There is exactly one Z2Z4 -cyclic 1-perfect code

AU - Borges, Joaquim

AU - Fernández-Córdoba, Cristina

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N2 - © 2016, Springer Science+Business Media New York. Let C be a Z2Z4-additive code of length n> 3. We prove that if the binary Gray image of C is a 1-perfect nonlinear code, then C cannot be a Z2Z4-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 Z2Z4-additive 1-perfect code gives a Z2Z4-additive extended 1-perfect code. We also prove that such a code cannot be Z2Z4-cyclic.

AB - © 2016, Springer Science+Business Media New York. Let C be a Z2Z4-additive code of length n> 3. We prove that if the binary Gray image of C is a 1-perfect nonlinear code, then C cannot be a Z2Z4-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 Z2Z4-additive 1-perfect code gives a Z2Z4-additive extended 1-perfect code. We also prove that such a code cannot be Z2Z4-cyclic.

KW - Perfect codes

KW - Simplex codes

KW - Z Z -additive cyclic codes 2 4

U2 - 10.1007/s10623-016-0323-3

DO - 10.1007/s10623-016-0323-3

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