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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

  • Dipak K. Bhunia
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Abstract

The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family Ht1,t2,t3 of ℤ2ℤ4ℤ8-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the ℤ2ℤ4-linear and ℤ2s-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear ℤ2ℤ4-linear Hadamard code and for every nonlinear ℤ2s-linear Hadamard code H̅a1,…,as with s≥2. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in s. We also prove a descent theorem: if H̅a1,…,as is nonlinear and a1≥2, then the code punctured on one kernel block is the ℤ2s−1-linear Hadamard code H̅a1,…,as−1. Consequently, the constant local rank equals rank(H̅a1,…,as−1) and is at least t−κ+2, where 2t is the length and κ the kernel dimension. These results separate every nonlinear ℤ2ℤ4ℤ8-linear Hadamard code from every ℤ4-linear, ℤ2ℤ4-linear and ℤ2s-linear Hadamard code of the same length, except for the single infinite family H1,1,t−4 and H̅2,0,t−5 with t≥5. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.

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Cite

BibTeXRIS
BibTeX
@misc{bhunia2026kernelblock,
  title = {{The kernel-block rank profiles of the $\mathbbZ_2\mathbbZ_4$-linear and the $\mathbbZ_2^s$-linear Hadamard codes, and a complete classification of the $\mathbbZ_2\mathbbZ_4\mathbbZ_8$-linear Hadamard codes}},
  author = {Dipak K. Bhunia},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00142},
  eprint = {2610.00142},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00142},
}
RIS
TY  - UNPB
TI  - The kernel-block rank profiles of the ℤ_2ℤ_4-linear and the ℤ_2^s-linear Hadamard codes, and a complete classification of the ℤ_2ℤ_4ℤ_8-linear Hadamard codes
AU  - Dipak K. Bhunia
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.00142
UR  - https://arxiv.org/abs/2610.00142
AB  - The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family H^t_1,t_2,t_3 of ℤ_2ℤ_4ℤ_8-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the ℤ_2ℤ_4-linear and ℤ_2^s-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear ℤ_2ℤ_4-linear Hadamard code and for every nonlinear ℤ_2^s-linear Hadamard code H̅^a_1,…,a_s with s≥2. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in s. We also prove a descent theorem: if H̅^a_1,…,a_s is nonlinear and a_1≥2, then the code punctured on one kernel block is the ℤ_2^s−1-linear Hadamard code H̅^a_1,…,a_s−1. Consequently, the constant local rank equals rank(H̅^a_1,…,a_s−1) and is at least t−κ+2, where 2^t is the length and κ the kernel dimension. These results separate every nonlinear ℤ_2ℤ_4ℤ_8-linear Hadamard code from every ℤ_4-linear, ℤ_2ℤ_4-linear and ℤ_2^s-linear Hadamard code of the same length, except for the single infinite family H^1,1,t−4 and H̅^2,0,t−5 with t≥5. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.
ER  -