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arXiv 1709.07586math.CO
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非交换群中的几乎差集

Almost Difference Sets in Nonabelian Groups

Jerod Michel, Qi Wang

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摘要：

给出了几乎差集的两种新构造。第一种是在以（$\mathbb{F}_{q},+)$）为子群的$q^{2}(q+1)$阶群（$q$是奇素数幂）中构造$(q^{2}(q+1),q(q^{2}-1),q(q^{2}-q-1),q^{2}-1)$几乎差集的一般方法。这种构造出现在任何以（$\mathbb{F}_{p^{2}},+)$）为加法子群的$p^{2}(p+1)$阶群（$p$是奇素数）中。这种构造产生了几个几乎差集的无限族，其中许多出现在非交换群中。第二种构造方法在$4p$阶二面体群中得到$(4p,2p+1,p,p-1)$几乎差集，其中$p\equiv 3 \ ({\rm mod} \ 4)$是素数。此外，还证明了在所得到的几乎差集的无限族中，有一些Cayley图是Ramanujan图. \keywords{Difference set \and Almost difference set \and Nonabelian group}

英文摘要：

We give two new constructions of almost difference sets. The first is a generic construction of $(q^{2}(q+1),q(q^{2}-1),q(q^{2}-q-1),q^{2}-1)$ almost difference sets in certain groups of order $q^{2}(q+1)$ ($q$ is an odd prime power) having ($\mathbb{F}_{q},+)$ as a subgroup. The construction occurs in any group of order $p^{2}(p+1)$ ($p$ is an odd prime) having ($\mathbb{F}_{p^{2}},+)$ as an additive subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields $(4p,2p+1,p,p-1)$ almost difference sets in dihedral groups of order $4p$ where $p\equiv 3 \ ({\rm mod} \ 4)$ is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained have Cayley graphs which are Ramanujan graphs. \keywords{Difference set \and Almost difference set \and Nonabelian group}

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