Linearity and classification of ZpZp2-linear generalized Hadamard codes

Abstract

The ZpZp2-additive codes are subgroups of Zpα1×Zp2α2, and can be seen as linear codes over Zp when α2=0, Zp2-additive codes when α1=0, or Z2Z4-additive codes when p=2. A ZpZp2-linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2-additive code. Recursive constructions of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1 are known. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0, and then we compare them with the ones obtained when α1=0. First, we show for which types the corresponding ZpZp2-linear GH codes are nonlinear over Zp. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike Z4-linear Hadamard codes, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0 when p≥3 prime. Indeed, there are some families with infinite nonlinear ZpZp2-linear GH codes, where the codes are not equivalent to any Zps-linear GH code with s≥2.