Construction and Linearity of Z_pZ_{p^2}-Linear Generalized Hadamard Codes

TL;DR

This paper extends the theory of $ ext{Z}_p ext{Z}_{p^2}$-linear generalized Hadamard codes from the case $p=2$ to any prime $p extgreater{}=3$, providing recursive constructions and analyzing non-linearity conditions.

Contribution

It introduces a recursive construction method for $ ext{Z}_p ext{Z}_{p^2}$-additive GH codes for any prime $p extgreater{}=3$ and characterizes when these codes are non-linear over $ ext{Z}_p$.

Findings

Recursive construction of $ ext{Z}_p ext{Z}_{p^2}$-additive GH codes.

Identification of types leading to non-linear GH codes.

$ ext{Z}_{p^2}$-linear GH codes are not contained in $ ext{Z}_p ext{Z}_{p^2}$-linear GH codes for $p extgreater{}=3$.

Abstract

The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{\alpha_1} \times \Z_{p^2}^{\alpha_2}$, and can be seen as linear codes over $\Z_p$ when $\alpha_2=0$, $\Z_{p^2}$-additive codes when $\alpha_1=0$, or $\Z_2\Z_4$-additive codes when $p=2$. A $\Z_p\Z_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the Gray map image of a $\Z_p\Z_{p^2}$-additive code. In this paper, we generalize some known results for $\Z_p\Z_{p^2}$-linear GH codes with $p=2$ to any $p\geq 3$ prime when $\alpha_1 \neq 0$. First, we give a recursive construction of $\Z_p\Z_{p^2}$-additive GH codes of type $(\alpha_1,\alpha_2;t_1,t_2)$ with $t_1,t_2\geq 1$. Then, we show for which types the corresponding $\Z_p\Z_{p^2}$-linear GH codes are non-linear over $\Z_p$. Finally, according to some computational results, we see that, unlike $\Z_4$-linear GH codes, when $p\geq 3$ prime, the $\Z_{p^2}$-linear GH codes are not included in the family of $\Z_p\Z_{p^2}$-linear GH codes with $\alpha_1\not =0$.