Full classification of ℤps-linear generalized Hadamard codes
Abstract
The ℤps-additive codes are subgroups of ℤpsn. A ℤps-linear generalized Hadamard (GH) code is a GH code over ℤp which is the Gray map image of a ℤps-additive code. A recursive construction gives such codes Hpt1,…,ts, of type (n;t1,…,ts) and length pt, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime p. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code Hpt1,…,ts with t1≥2, the derived code is a ℤps−1-linear GH code of the same family, although the mechanism for odd p differs from the binary one. As a consequence, for t≥3, there are 𝒫(t+1)−𝒫(t) nonequivalent such codes of length pt if p is odd, and 𝒫(t+1)−𝒫(t)−1 if p=2, where 𝒫(n) is the number of partitions of n. In particular, this completes the classification of the ℤ2s-linear Hadamard codes. We also prove a conjecture on the codes with t1≥2 and ts=0.
Related
- 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
- Nonequivalence of ℤpℤp2-linear and ℤps-linear generalized Hadamard codes for odd primes
- Rank and classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- The kernel-block rank profile and a complete classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- The rank of ℤ2s-linear Hadamard codes
BibTeX
@misc{bhunia2026full,
title = {{Full classification of $\mathbbZ_p^s$-linear generalized Hadamard codes}},
author = {Dipak K. Bhunia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18409},
eprint = {2610.18409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18409},
}
RIS
TY - UNPB
TI - Full classification of ℤ_p^s-linear generalized Hadamard codes
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18409
UR - https://arxiv.org/abs/2610.18409
AB - The ℤ_p^s-additive codes are subgroups of ℤ_p^s^n. A ℤ_p^s-linear generalized Hadamard (GH) code is a GH code over ℤ_p which is the Gray map image of a ℤ_p^s-additive code. A recursive construction gives such codes H_p^t_1,…,t_s, of type (n;t_1,…,t_s) and length p^t, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime p. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code H_p^t_1,…,t_s with t_1≥2, the derived code is a ℤ_p^s−1-linear GH code of the same family, although the mechanism for odd p differs from the binary one. As a consequence, for t≥3, there are 𝒫(t+1)−𝒫(t) nonequivalent such codes of length p^t if p is odd, and 𝒫(t+1)−𝒫(t)−1 if p=2, where 𝒫(n) is the number of partitions of n. In particular, this completes the classification of the ℤ_2^s-linear Hadamard codes. We also prove a conjecture on the codes with t_1≥2 and t_s=0.
ER -