TY - JOUR
T1 - On recursive constructions of Z_2Z_4Z_8-linear Hadamard codes
AU - Bhunia, Dipak K.
AU - Department of Information and Communications Engineering, Universitat Autònoma de Barcelona
AU - Fernández-Córdoba, Cristina
AU - Villanueva, Mercè
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
UR - https://www.mendeley.com/catalogue/4850b1cf-9325-32f4-b81c-d54e6fb48cbf/
U2 - 10.3934/amc.2023047
DO - 10.3934/amc.2023047
M3 - Article
SN - 1930-5346
JO - Advances in Mathematics of Communications
JF - Advances in Mathematics of Communications
ER -