A $G$-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes

2026-10-07

AI Summary

This study addresses the existence determination of asymptotically optimal free invariant packings under permutation group actions and the bounding of optical codes. By integrating combinatorics, group theory, and asymptotic analysis, this work proposes a novel theoretical framework for group-invariant packings, employing constructive proofs grounded in algebraic structural properties. The primary contributions include establishing a G-invariant Erdős–Hanani theorem and deriving necessary and sufficient conditions for the existence of asymptotically optimal free t-packings under semiregular groups, thereby transcending the symmetry constraints inherent in traditional combinatorial designs. Furthermore, it confirms the asymptotic attainability of the Johnson bound for optical orthogonal codes and constant-weight codes, while precisely determining the maximum size of two-dimensional optical codes.

Abstract

Let $k>t\ge 2$ 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 $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ 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 $t\ge 3$, 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 $w\geλ+2$, 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.

Problem

• G-invariant packing
• Erdős-Hanani theorem
• optical orthogonal codes
• constant-weight codes
• Johnson bound

Innovation

• G-invariant packing
• Erdős-Hanani theorem
• optical orthogonal codes
• constant-weight codes
• semiregular groups