On the linearity and classification of Z_p^s -linear generalized Hadamard codes

Zp^s-additive codes of length n are subgroups of (Zp^s)^n , and can be seen as a generalization of linear codes over Z2, Z4 , or Z2^s in general. A Zp^s-linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zp^s-additive code by a generalized Gray map. In this paper, we generalize some known results for Zp^s-linear GH codes with p = 2 to any odd prime p. First, we show some results related to the generalized Carlet’s Gray map. Then, by using an iterative construction of Zp^s -additive GH codes of type (n; t 1 , . . . , t s ), we show for which types the corresponding Zp^s-linear GH codes of length p^t are nonlinear over Zp .For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for p ≥ 3 are different from the case with p = 2. Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any p ≥ 2; by using also the rank as an invariant in some specific cases.