On the equivalence of $$\mathbb {Z}_{p^s}$$-linear generalized Hadamard codes

Abstract

AbstractLinear codes of length n over $$\mathbb {Z}_{p^s}$$ Z p s , p prime, called $$\mathbb {Z}_{p^s}$$ Z p s -additive codes, can be seen as subgroups of $$\mathbb {Z}_{p^s}^n$$ Z p s n . 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 under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for $$p\ge 3$$ p ≥ 3 prime, we establish that some $$\mathbb {Z}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to $$t=10$$ t = 10 if $$p=3$$ p = 3 or $$t=8$$ t = 8 if $$p=5$$ p = 5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel.