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The rank of ℤ2s-linear Hadamard codes
Abstract
The canonical ℤ2s-linear Hadamard codes are the Gray images of the ℤ2s-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for s≤ 3, and for s ≥ 4 only through computer computations for small lengths. We give a method that computes the rank for every s. We show that the rank is a binomial polynomial in the type parameters, with coefficients ρs(k) independent of the length. We compute all the coefficients for s≤ 5, which yields explicit rank formulas over ℤ4 and ℤ8 (recovering the known ones), and over ℤ16 and ℤ32.
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
- Rank and classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- The kernel-block rank profile and a complete classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- Full classification of ℤps-linear generalized Hadamard codes
- Nonequivalence of ℤpℤp2-linear and ℤps-linear generalized Hadamard codes for odd primes
BibTeX
@misc{wang2026rank,
title = {{The rank of $\mathbbZ_2^s$-linear Hadamard codes}},
author = {Xuan Wang and Minjia Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34730},
eprint = {2610.34730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34730},
}
RIS
TY - UNPB
TI - The rank of ℤ_2^s-linear Hadamard codes
AU - Xuan Wang
AU - Minjia Shi
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34730
UR - https://arxiv.org/abs/2610.34730
AB - The canonical ℤ_2^s-linear Hadamard codes are the Gray images of the ℤ_2^s-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for s≤ 3, and for s ≥ 4 only through computer computations for small lengths. We give a method that computes the rank for every s. We show that the rank is a binomial polynomial in the type parameters, with coefficients ρ_s(k) independent of the length. We compute all the coefficients for s≤ 5, which yields explicit rank formulas over ℤ_4 and ℤ_8 (recovering the known ones), and over ℤ_16 and ℤ_32.
ER -