On the constructions of [formula omitted]-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. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0. First, we give a recursive construction of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1. We also present many different recursive constructions of ZpZp2-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z4-linear GH codes, when p≥3 prime, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0. Indeed, we observe that the constructed codes are not equivalent to the Zps-linear GH codes for any s≥2.