Geometric construction of association schemes from degenerate quadrics in projective spaces PG(2ν+δ+l-1, Fq) for q odd

Let F-q be a finite field with q elements, where q is a power of an odd prime. In this paper, the authors consider a projective space PG(2 nu + delta + iota, F-q) with dimension 2 nu + delta + iota, partitioned into an affine space AG(2 nu + delta + iota, F-q) of dimension 2 nu + delta + iota and a hyperplane H = PG(2 nu + delta + iota -1, F-q) of dimension 2 nu + delta + iota - 1 at infinity, where iota not equal 0. The points of the hyperplane H are next partitioned into four subsets. A pair of points a. and b of the affine space is defined to belong to class i if the line <(ab)over bar> meets the subset i of H. Finally, a family of four-class association schemes are constructed, and parameters are also computed.