Sets with integral distances in finite fields

Abstract

Given a positive integer n n , a finite field F q \mathbb F_q of q q elements ( q q odd), and a non-degenerate quadratic form Q Q on F q n \mathbb {F}_q^n , in this paper we study the largest possible cardinality of subsets E ⊆ F q n \mathcal {E} \subseteq \mathbb {F}_q^n with pairwise integral Q Q -distances; that is, for any two vectors {x} = ( x 1 , … , x n ) , {y} = ( y 1 , … , y n ) ∈ E \textbf {{x}}=(x_1, \ldots ,x_n), \textbf {{y}}=(y_1,\ldots ,y_n) \in \mathcal {E} , one has \[ Q ( {x} − {y} ) = u 2 Q(\textbf {{x}}-\textbf {{y}})=u^2 \] for some u ∈ F q u \in \mathbb F_q .