Nonequivalence of ℤpℤp2-linear and ℤps-linear generalized Hadamard codes for odd primes
Abstract
The ℤpℤp2-additive codes are subgroups of ℤpα1×ℤp2α2, and they can be seen as linear codes over ℤp when α2=0, as ℤp2-additive codes when α1=0, and as 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. For every prime p, there are recursive constructions of ℤpℤp2-linear GH codes Hpt1,t2 with α1≠0 and α2≠0, and of ℤps-linear GH codes H̅pt1,…,ts. We show that the kernel blocks of a nonlinear code Hpt1,t2 give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of Hpt1,t2. We also determine the ranks of the codes Hpt1,t2 and H̅pt1,t2. For odd p, the rank of Hpt1,t2 depends on the Bernoulli numbers modulo p. By comparing the derived codes and the ranks, we show that, for odd p, no code Hpt1,t2 is equivalent to any of these ℤps-linear GH codes. For p=2, a nonlinear code H2t1,t2 is equivalent to a ℤ2s-linear Hadamard code if and only if t2≥2. Finally, we count the nonequivalent codes in the union of both families.
Related
- Full classification of ℤps-linear generalized Hadamard codes
- The rank of ℤ2s-linear Hadamard codes
- The kernel-block rank profiles of the ℤ2ℤ4-linear and the ℤ2s-linear Hadamard codes, and a complete classification of the ℤ2ℤ4ℤ8-linear Hadamard codes
- Rank and classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- The kernel-block rank profile and a complete classification of ℤ2ℤ4ℤ8-linear Hadamard codes
BibTeX
@misc{bhunia2026nonequivalence,
title = {{Nonequivalence of $\mathbbZ_p\mathbbZ_p^2$-linear and $\mathbbZ_p^s$-linear generalized Hadamard codes for odd primes}},
author = {Dipak K. Bhunia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28306},
eprint = {2610.28306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28306},
}
RIS
TY - UNPB
TI - Nonequivalence of ℤ_pℤ_p^2-linear and ℤ_p^s-linear generalized Hadamard codes for odd primes
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28306
UR - https://arxiv.org/abs/2610.28306
AB - The ℤ_pℤ_p^2-additive codes are subgroups of ℤ_p^α_1×ℤ_p^2^α_2, and they can be seen as linear codes over ℤ_p when α_2=0, as ℤ_p^2-additive codes when α_1=0, and as ℤ_2ℤ_4-additive codes when p=2. A ℤ_pℤ_p^2-linear generalized Hadamard (GH) code is a GH code over ℤ_p which is the Gray map image of a ℤ_pℤ_p^2-additive code. For every prime p, there are recursive constructions of ℤ_pℤ_p^2-linear GH codes H_p^t_1,t_2 with α_1≠0 and α_2≠0, and of ℤ_p^s-linear GH codes H̅_p^t_1,…,t_s. We show that the kernel blocks of a nonlinear code H_p^t_1,t_2 give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of H_p^t_1,t_2. We also determine the ranks of the codes H_p^t_1,t_2 and H̅_p^t_1,t_2. For odd p, the rank of H_p^t_1,t_2 depends on the Bernoulli numbers modulo p. By comparing the derived codes and the ranks, we show that, for odd p, no code H_p^t_1,t_2 is equivalent to any of these ℤ_p^s-linear GH codes. For p=2, a nonlinear code H_2^t_1,t_2 is equivalent to a ℤ_2^s-linear Hadamard code if and only if t_2≥2. Finally, we count the nonequivalent codes in the union of both families.
ER -