Sets with integral distances in finite fields

Abstract:

Given a positive integer $n$, a finite field $\mathbb F_q$ of $q$ elements ($q$ odd), and a non-degenerate quadratic form $Q$ on $\mathbb {F}_q^n$, in this paper we study the largest possible cardinality of subsets $\mathcal {E} \subseteq \mathbb {F}_q^n$ with pairwise integral $Q$-distances; that is, for any two vectors $\textbf {{x}}=(x_1, \ldots ,x_n), \textbf {{y}}=(y_1,\ldots ,y_n) \in \mathcal {E}$, one has \[ Q(\textbf {{x}}-\textbf {{y}})=u^2\] for some $u \in \mathbb F_q$.