The kernel-block rank profile and a complete classification of ℤ2ℤ4ℤ8-linear Hadamard codes
Abstract
The ℤ2ℤ4ℤ8-additive codes are subgroups of ℤ2α1×ℤ4α2×ℤ8α3, and a ℤ2ℤ4ℤ8-linear Hadamard code is the Gray map image of such a code. A recursive construction of ℤ2ℤ4ℤ8-additive Hadamard codes ℋt1,t2,t3, with all αi≠0, t1≥1, t2≥0, and t3≥1, is known, as are the linearity, kernel dimension, and rank of the corresponding codes Ht1,t2,t3 of length 2t, where t+1=3t1+2t2+t3. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths 2t, 3≤ t≤11, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are 2t1+t2+t3−1 blocks, all of size 22t1+t2, and the profile takes at most two values t2+binomt1+22 and t2+2+binomt1+12, whose difference is t1−1; it is constant precisely when t1=1. Hence, the profile recovers t1, and the length and kernel dimension recover t2 and t3. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length 2t is ⌊(t2+6)/12⌋ for every t≥3.
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
- A Complete Classification of Complex Hadamard Matrices of Order Six
- Full classification of ℤps-linear generalized Hadamard codes
- Rank and classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- A complete classification of the existence of finite-dimensional quantum solutions to inconsistent linear constraint systems
BibTeX
@misc{bhunia2026kernelblock,
title = {{The kernel-block rank profile and a complete classification of $\mathbbZ_2\mathbbZ_4\mathbbZ_8$-linear Hadamard codes}},
author = {Dipak K. Bhunia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09969},
eprint = {2609.09969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09969},
}
RIS
TY - UNPB
TI - The kernel-block rank profile and a complete classification of ℤ_2ℤ_4ℤ_8-linear Hadamard codes
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09969
UR - https://arxiv.org/abs/2609.09969
AB - The ℤ_2ℤ_4ℤ_8-additive codes are subgroups of ℤ_2^α_1×ℤ_4^α_2×ℤ_8^α_3, and a ℤ_2ℤ_4ℤ_8-linear Hadamard code is the Gray map image of such a code. A recursive construction of ℤ_2ℤ_4ℤ_8-additive Hadamard codes ℋ^t_1,t_2,t_3, with all α_i≠0, t_1≥1, t_2≥0, and t_3≥1, is known, as are the linearity, kernel dimension, and rank of the corresponding codes H^t_1,t_2,t_3 of length 2^t, where t+1=3t_1+2t_2+t_3. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths 2^t, 3≤ t≤11, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are 2^t_1+t_2+t_3−1 blocks, all of size 2^2t_1+t_2, and the profile takes at most two values t_2+binomt_1+22 and t_2+2+binomt_1+12, whose difference is t_1−1; it is constant precisely when t_1=1. Hence, the profile recovers t_1, and the length and kernel dimension recover t_2 and t_3. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length 2^t is ⌊(t^2+6)/12⌋ for every t≥3.
ER -