Sets with many pairs of orthogonal vectors over finite fields

Let n   be a positive integer and BB be a non-degenerate symmetric bilinear form over Fqn, where q   is an odd prime power and FqFq 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|{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)=0B(x,y)=0.