On recursive constructions of Z_2Z_4Z_8-linear Hadamard codes

Dipak K. Bhunia, Universitat Autònoma de Barcelona Department of Information and Communications Engineering, Cristina Fernández-Córdoba, Mercè Villanueva

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Resum

The Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0, α2 ̸= 0, and α3 ̸= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 ̸= 0 and α2 ̸= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s ≥ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.
Idioma originalAnglès
Nombre de pàgines25
RevistaAdvances in Mathematics of Communications
DOIs
Estat de la publicacióPublicada - 2023

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