ℤ_2ℤ_4ℤ_8-Additive Hadamard Codes
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The ℤ_2ℤ_4ℤ_8-additive codes are subgroups of ℤ_2^α_1×ℤ_4^α_2×ℤ_8^α_3, and can be seen as linear codes over ℤ_2 when α_2=α_3=0, ℤ_4-additive or ℤ_8-additive codes when α_1=α_3=0 or α_1=α_2=0, respectively, or ℤ_2ℤ_4-additive codes when α_3=0. A ℤ_2ℤ_4ℤ_8-linear Hadamard code is a Hadamard code which is the Gray map image of a ℤ_2ℤ_4ℤ_8-additive code. In this paper, we generalize some known results for ℤ_2ℤ_4-linear Hadamard codes to ℤ_2ℤ_4ℤ_8-linear Hadamard codes with α_1 ≠ 0, α_2 ≠ 0, and α_3 ≠ 0. First, we give a recursive construction of ℤ_2ℤ_4ℤ_8-additive Hadamard codes of type (α_1,α_2, α_3;t_1,t_2, t_3) with t_1≥ 1, t_2 ≥ 0, and t_3≥ 1. Then, we show that in general the ℤ_4-linear, ℤ_8-linear and ℤ_2ℤ_4-linear Hadamard codes are not included in the family of ℤ_2ℤ_4ℤ_8-linear Hadamard codes with α_1 ≠ 0, α_2 ≠ 0, and α_3 ≠ 0. Actually, we point out that none of these nonlinear ℤ_2ℤ_4ℤ_8-linear Hadamard codes of length 2^11 is equivalent to a ℤ_2ℤ_4ℤ_8-linear Hadamard code of any other type, a ℤ_2ℤ_4-linear Hadamard code, or a ℤ_2^s-linear Hadamard code, with s≥ 2, of the same length 2^11.
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