Sets with many pairs of orthogonal vectors over finite fields

Abstract

Let n be a positive integer and B be a non-degenerate symmetric bilinear form over Fqn, where q is an odd prime power and Fq is the finite field with q elements. We determine the largest possible size of a subset S of Fqn such that |{B(x,y)|x,y∈S and x≠y}|=1. We also pose some conjectures concerning nearly orthogonal subsets of Fqn where a nearly orthogonal subset T of Fqn is a set of vectors in which among any three distinct vectors there are two vectors x, y so that B(x,y)=0.