2023/01/23 by Bhunia, Dipak K., Fernández-Córdoba, Cristina, Villanueva, Mercè
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2301.09404
The ℤ2ℤ4ℤ8-additive codes are subgroups of ℤ2α1 × ℤ4α2 × ℤ8α3, and can be seen as linear codes over ℤ2 when α2=α3=0, ℤ4-additive or ℤ8-additive codes when α1=α3=0 or α1=α2=0, respectively, or ℤ2ℤ4-additive codes when α3=0. 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. Then, we show that in general the ℤ4-linear, ℤ8-linear and ℤ2ℤ4-linear Hadamard codes are not included in the family of ℤ2ℤ4ℤ8-linear Hadamard codes with α1 ≠ 0, α2 ≠ 0, and α3 ≠ 0. Actually, we point out that none of these nonlinear ℤ2ℤ4ℤ8-linear Hadamard codes of length 211 is equivalent to a ℤ2ℤ4ℤ8-linear Hadamard code of any other type, a ℤ2ℤ4-linear Hadamard code, or a ℤ2s-linear Hadamard code, with s≥ 2, of the same length 211.