Rank and classification of ℤ2ℤ4ℤ8-linear Hadamard codes
Abstract
The ℤ2ℤ4ℤ8-additive codes are subgroups of ℤ2α1×ℤ4α2×ℤ8α3. A ℤ2ℤ4ℤ8-linear Hadamard code is a Hadamard code which 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 and kernel of the corresponding codes Ht1,t2,t3 of length 2t. Their rank, however, was not known in closed form. In this paper, we determine rank(Ht1,t2,t3) for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are (H1,2,t−6,H2,0,t−5) for t≥7 and (H2,2,t−9,H3,0,t−8) for t≥10. Consequently, the number of distinct pairs (r,k) among the codes of length 2t is ⌊(t2+6)/12⌋ for 3≤ t≤6, ⌊(t2+6)/12⌋−1 for 7≤ t≤9, and ⌊(t2+6)/12⌋−2 for t≥10. Thus, rank and kernel dimension classify the family completely if and only if 3≤ t≤6. We also determine all coincidences with the ℤ4-, ℤ2ℤ4-, and ℤ8-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely H2,1,t−7 and H5,t−9 for t≥10, and exactly five families with the ℤ8-linear codes.
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
- The kernel-block rank profile and a complete classification of ℤ2ℤ4ℤ8-linear Hadamard codes
- The rank of ℤ2s-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{bhunia2026rank,
title = {{Rank and 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.2609.06244},
eprint = {2609.06244},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06244},
}
RIS
TY - UNPB
TI - Rank and classification of ℤ_2ℤ_4ℤ_8-linear Hadamard codes
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06244
UR - https://arxiv.org/abs/2609.06244
AB - The ℤ_2ℤ_4ℤ_8-additive codes are subgroups of ℤ_2^α_1×ℤ_4^α_2×ℤ_8^α_3. A ℤ_2ℤ_4ℤ_8-linear Hadamard code is a Hadamard code which 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 and kernel of the corresponding codes H^t_1,t_2,t_3 of length 2^t. Their rank, however, was not known in closed form. In this paper, we determine rank(H^t_1,t_2,t_3) for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are (H^1,2,t−6,H^2,0,t−5) for t≥7 and (H^2,2,t−9,H^3,0,t−8) for t≥10. Consequently, the number of distinct pairs (r,k) among the codes of length 2^t is ⌊(t^2+6)/12⌋ for 3≤ t≤6, ⌊(t^2+6)/12⌋−1 for 7≤ t≤9, and ⌊(t^2+6)/12⌋−2 for t≥10. Thus, rank and kernel dimension classify the family completely if and only if 3≤ t≤6. We also determine all coincidences with the ℤ_4-, ℤ_2ℤ_4-, and ℤ_8-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely H^2,1,t−7 and H^5,t−9 for t≥10, and exactly five families with the ℤ_8-linear codes.
ER -