On Cross Parsons Numbers

Abstract

Let $F_q$ be the field of size q and SL(n, q) be the special linear group of order n over the field $F_q$. Assume that n is an even integer. Let ${\mathcal {A}}_i\subseteq SL(n,q)$ for $i=1,2,\dots , k$ and $\vert {\mathcal {A}}_1\vert =\vert \mathcal A_2\vert =\cdots =\vert {\mathcal {A}}_k\vert =l$. The set $\{\mathcal A_1,{\mathcal {A}}_2,\dots ,{\mathcal {A}}_k\}$ is called a k-cross (n, q)-Parsons set of size l, if for any pair of (i, j) with $i\ne j$, $A-B\in SL(n,q)$ for all $A\in {\mathcal {A}}_i$ and $B\in {\mathcal {A}}_j$. Let m(k, n, q) be the largest integer l for which there is a k-cross (n, q)-Parsons set of size l. The integer m(k, n, q) will be called the k-cross (n, q)-Parsons numbers. In this paper, we will show that $m(3,2,q)\le q$. Furthermore, $m(3,2,q)= q$ if and only if $q=4^r$ for some positive integer r. We will also show that if n is a multiple of $q-1$, then $m(q-1,n,q)\ge q^{\frac{1}{2}n(n-1)}$.