Sets with many pairs of orthogonal vectors over finite fields

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

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