发表机构
Singapore University of Technology and Design（新加坡科技设计大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出G-不变的Erdős–Hanani定理，证明半正则群满足相关条件时存在渐近最优自由G-不变填充，将其应用于光正交码、常重码等，确定了二维光正交码的渐近最大规模。
AI 中文摘要
设k>t≥2为固定整数，G为v个点集合上的置换群。G-不变的填充是自由的，当且仅当每个区组在G作用下有|G|个不同的像。本文给出G满足的两个条件：当v→∞时，存在自由的G-不变t-(v,k,1)填充，其区组数为(1-o(1))binom{v}{t}/binom{k}{t}，这是渐近最优填充的Erdős–Hanani定理的G-不变形式。对于半正则群G，具有这些性质的填充存在当且仅当除了o(v^t)个t-子集外，其余所有t-子集都具有平凡的集合稳定子；该性质对t≥3时的所有半正则群成立，对t=2时的所有循环半正则群成立。由此可得，当w≥λ+2时，常重w、常相关λ的光正交码及其多维、签名模式版本，以及循环和准循环常重码，在所有长度下都渐近达到Johnson界。相同方法给出了循环群不变的可分组填充的定理形式，确定了每个波长最多一个脉冲的二维光正交码的渐近最大规模。
英文摘要
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.
Comments16 pages, 1 table
