On the Classification of ℤpℤp2-Linear Generalized Hadamard Codes
Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva · 2022 IEEE Information Theory Workshop (ITW) · 2022
The ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}}$ codes are subgroups of $\mathbb{Z}_p^{{\alpha _1}} \times \mathbb{Z}_{{p^2}}^{{\alpha _2}}$. A${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}}$ generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}_p}$ which is the Gray map image of a ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-additive code. A recursive construction of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}}$ GH codes of type (α1, α2; t1, t2) with t1, t2≥ 1 is known, and for which types the corresponding ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-linear GH codes are nonlinear over ${\mathbb{Z}_p}$ is also known. In this paper, we generalize some known results for ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-linear GH codes with p = 2 to any p≥3 prime when ${\alpha _1} e 0$. First, we present new recursive constructions of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}}$ GH codes having the same type, and show that we obtained equivalent codes. Then, we compute the rank of some families of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes. Finally, we show that, unlike ${\mathbb{Z}_4}{\text{ - linear}}$ Hadamard codes, the ${\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes are not included in the family of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes with ${\alpha _1} e 0$ when p ≥ 3 prime.