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

Abstract

Abstract$${\mathbb {Z}}_{p^s}$$ Z p s -additive codes of length n are subgroups of $${\mathbb {Z}}_{p^s}^n$$ Z p s n , and can be seen as a generalization of linear codes over $${\mathbb {Z}}_2$$ Z 2 , $${\mathbb {Z}}_4$$ Z 4 , or $${\mathbb {Z}}_{2^s}$$ Z 2 s in general. A $${\mathbb {Z}}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $${\mathbb {Z}}_p$$ Z p which is the image of a $${\mathbb {Z}}_{p^s}$$ Z p s -additive code by a generalized Gray map. In this paper, we generalize some known results for $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes with $$p=2$$ 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 $${\mathbb {Z}}_{p^s}$$ Z p s -additive GH codes of type $$(n;t_1,\ldots , t_s)$$ ( n ; t 1 , … , t s ) , we show for which types the corresponding $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t are nonlinear over $${\mathbb {Z}}_p$$ Z p . For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for $$p\ge 3$$ p ≥ 3 are different from the case with $$p=2$$ p = 2 . Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any $$p\ge 2$$ p ≥ 2 ; by using also the rank as an invariant in some specific cases.