ℤ_pℤ_p^2…ℤ_p^s-Additive Generalized Hadamard Codes
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The ℤ_pℤ_p^2…ℤ_p^s-additive codes are subgroups of ℤ_p^α_1×ℤ_p^2^α_2×⋯×ℤ_p^s^α_s, and can be seen as linear codes over ℤ_p when α_i=0 for all i ∈{2,3,…, s}, a ℤ_p^s-additive code when α_i=0 for all i ∈{1,2,…, s-1} , or a ℤ_pℤ_p^2-additive code when s=2, ℤ_2ℤ_4-additive codes when p=2 and s=2. A ℤ_pℤ_p^2…ℤ_p^s-linear generalized Hadamard (GH) code is a GH code over ℤ_p which is the Gray map image of a ℤ_pℤ_p^2…ℤ_p^s-additive code. In this paper, we generalize some known results for ℤ_pℤ_p^2…ℤ_p^s-linear GH codes with p prime and s≥ 2. First, we give a recursive construction of ℤ_pℤ_p^2…ℤ_p^s-additive GH codes of type (α_1,…,α_s;t_1,…,t_s) with t_1≥ 1, t_2,…,t_s-1≥ 0 and t_s≥1. Then, we show for which types the corresponding ℤ_pℤ_p^2…ℤ_p^s-linear GH codes are nonlinear over ℤ_p. We also compute the kernel and its dimension whenever they are nonlinear.
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