ON RECURSIVE CONSTRUCTIONS OF Z2Z4Z8-LINEAR HADAMARD CODES

Abstract

The Z2Z4Z8-additive codes are subgroups of (Formula present)-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.