Paley图和Peisert图的设计
Designs from Paley graphs and Peisert graphs
摘要：
固定正整数$p,q,$和$r$，使$p$为素数，$q=p^r$和$q\equiv 1$（mod $4$）。如下固定图$G$：如果$r$是奇数或$p\not\equiv 3$（mod $4$），令$G$是$q$ -顶点Paley图;如果$r$是偶数且$p\equiv 3$（mod $4$），令$G$是$q$ -顶点Paley图或$q$ -顶点Peisert图。我们利用$G$的子图结构构造了四个$2$ -设计序列，并计算了它们的参数。让$k_4$表示$G$中$4$ -顶点团的数目，我们从$G$中创建$2$ -设计的$62$附加序列，并展示如何仅用$q$和$k_4$来表示它们的参数.在$G$是Paley图的情况下，我们找到了$k_4$的估计和精确的渐近性。我们还解释了如何提出的技术可以用来找到许多额外的$2$设计$G$。所有构造的设计均不包含重复区组。
英文摘要：
Fix positive integers $p,q,$ and $r$ so that $p$ is prime, $q=p^r$, and $q\equiv 1$ (mod $4$). Fix a graph $G$ as follows: If $r$ is odd or $p\not\equiv 3$ (mod $4$), let $G$ be the $q$-vertex Paley graph; if $r$ is even and $p\equiv 3$ (mod $4$), let $G$ be either the $q$-vertex Paley graph or the $q$-vertex Peisert graph. We use the subgraph structure of $G$ to construct four sequences of $2$-designs, and we compute their parameters. Letting $k_4$ denote the number of $4$-vertex cliques in $G$, we create $62$ additional sequences of $2$-designs from $G$, and show how to express their parameters in terms of only $q$ and $k_4$. We find estimates and precise asymptotics for $k_4$ in the case that $G$ is a Paley graph. We also explain how the presented techniques can be used to find many additional $2$-designs in $G$. All constructed designs contain no repeated blocks.
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