相互正交的反拉丁方

中文摘要 AI 辅助

反拉丁方是在非线性安全网络编码相关研究中引入的，而该场景下对大型相互正交族的极值问题研究具有重要动机。本文研究了阶为d的相互正交反拉丁方族的最大规模$N_A(d)$，证明了对所有$d\geq3$，均有$N_L(d)+1\leq N_A(d)\leq N_L(d)+2$，其中$N_L(d)$表示阶为d的相互正交拉丁方族的经典最大规模；进一步得出，$N_A(3)=N_L(3)+1$，而对所有$d\geq4$，$N_A(d)=N_L(d)+2$。上界通过过渡到平衡矩阵获得，下界则由确定性置换论证给出。对所有$d\geq8$以及特殊阶$d=6$，通过一般概率构造证明上界可实现。结构方面，本文证明了大小为$d+1$的饱和族可诱导出阶为d的仿射平面，且饱和情形的特征是存在反坐标网格分解；将该条件迁移至固定单元集$[d]^2$后，对应行块与列块上会呈现方向完备性条件。其余小阶数单独处理：$d=3$通过正交三元组的直接分析与分类处理，$d=4$通过显式饱和构造及其有限几何结构分析处理，$d=5$和$d=7$则通过随机网格框架产生的显式饱和实例处理。由此，对所有$d\geq3$，$N_A(d)$可通过$N_L(d)$确定，且对所有$3\leq d\leq9$，其数值均可明确获取。

英文摘要

Anti-Latin squares were introduced in connection with non-linear secure network coding, and the extremal problem for large mutually orthogonal families is motivated by that setting. We study the maximum size $N_A(d)$ of a family of mutually orthogonal anti-Latin squares of order $d$. We prove that $N_L(d)+1\le N_A(d)\le N_L(d)+2$ for every $d\ge 3$, where $N_L(d)$ denotes the classical maximum size of a family of mutually orthogonal Latin squares of order $d$, and we show that in fact $N_A(3)=N_L(3)+1$ whereas $N_A(d)=N_L(d)+2$ for every $d\ge 4$. The upper bound is obtained by passing through balanced matrices, while the lower bound is given by a deterministic permutation argument. For all $d\ge 8$, and also for the exceptional order $d=6$, the upper bound is shown to be attainable by a general probabilistic construction. On the structural side, we show that a saturated family of size $d+1$ induces an affine plane of order $d$, and that the saturated case is characterized by the existence of an anti-coordinate grid decomposition; after transporting this condition to the fixed cell set $[d]^2$, it becomes a direction-completeness condition on the corresponding row-blocks and column-blocks. The remaining small orders are treated separately: $d=3$ is handled by direct analysis and classification of orthogonal triples, $d=4$ by an explicit saturated construction and an analysis of its finite-geometric structure, and $d=5$ and $d=7$ by explicit saturated examples arising from the random-grid framework. Thus $N_A(d)$ is determined in terms of $N_L(d)$ for every $d\ge3$, and its numerical value is obtained explicitly for every $3\le d\le9$.

发表机构

• Tokai Senior High School（东海高中）
• School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学深圳数据科学学院）
• Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）

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