On the linearity and classification of $${\mathbb {Z}}_{p^s}$$-linear generalized hadamard codes

Abstract $${\mathbb {Z}}_{p^s}$$ Z p s -additive codes of length n are subgroups {Z}}_{p^s}^n$$ n , and can be seen as a generalization linear over {Z}}_2$$ 2 {Z}}_4$$ 4 or {Z}}_{2^s}$$ in general. A -linear generalized Hadamard (GH) code is GH {Z}}_p$$ which the image by Gray map. In this paper, we generalize some known results for with $$p=2$$ = to any odd prime p . First, show related Carlet’s Then, using an iterative construction type $$(n;t_1,\ldots t_s)$$ ( ; t 1 , … ) types corresponding $$p^t$$ nonlinear For these codes, compute kernel its dimension, allow us give partial classification. The obtained $$p\ge 3$$ ≥ 3 different from case Finally, exact number non-equivalent such given infinite values s t 2$$ ; also rank invariant specific cases.