There is exactly one Z2Z4-cyclic 1-perfect code

2015/10/21 by Borges, Joaquim, Fernández-Córdoba, Cristina
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.1510.06166

Abstract

Let \cal C be a ℤ₂ℤ₄-additive code of length n > 3. We prove that if the binary Gray image of \cal C, C=Φ(\cal C), is a 1-perfect nonlinear code, then \cal 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 an extended 1-perfect code. We also prove that any such code cannot be ℤ₂ℤ₄-cyclic.

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