Geometric construction of association schemes from non-degenerate quadrics

LetFq be a finite field withq elements, whereq is a power of an odd prime. In this paper, we assume that δ=0,1 or 2 and consider a projective spacePG(2ν+δ,Fq), partitioned into an affine spaceAG(2ν+δ,Fq) of dimension 2ν+δ and a hyperplaneℋ=PG(2ν+δ−1,Fq) of dimension 2ν+δ−1 at infinity. The points of the hyperplaneℋ are next partitioned into three subsets. A pair of pointsa andb of the affine space is defined to belong to classi if the line\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {ab} $$ \end{document} meets the subseti of ℋ. Finally, we derive a family of three-class association schemes, and compute their parameters.