A G-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes
Abstract
Let k>t≥ 2 be fixed integers, and let G be a permutation group on a set of v points. A G-invariant packing is free if every block has |G| distinct images under G. We give two conditions on G under which, as v→∞, there is a free G-invariant t-(v,k,1) packing with (1−o(1))binomvt/binomkt blocks. This is a G-invariant form of the Erdős--Hanani theorem on asymptotically optimal packings. For semiregular groups G, packings with these properties exist if and only if all but o(vt) of the t-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for t≥ 3, and by every cyclic semiregular group for t=2. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight w and constant correlation λ with w≥λ+2, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.
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BibTeX
@misc{chee2026ginvariant,
title = {{A $G$-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes}},
author = {Yeow Meng Chee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10973},
eprint = {2610.10973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10973},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A G-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes
AU - Yeow Meng Chee
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10973
UR - https://arxiv.org/abs/2610.10973
AB - Let k>t≥ 2 be fixed integers, and let G be a permutation group on a set of v points. A G-invariant packing is free if every block has |G| distinct images under G. We give two conditions on G under which, as v→∞, there is a free G-invariant t-(v,k,1) packing with (1−o(1))binomvt/binomkt blocks. This is a G-invariant form of the Erdős--Hanani theorem on asymptotically optimal packings. For semiregular groups G, packings with these properties exist if and only if all but o(v^t) of the t-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for t≥ 3, and by every cyclic semiregular group for t=2. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight w and constant correlation λ with w≥λ+2, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.
ER -