Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

Abstract

Taking a nondegenerate Hermitian variety as a projective set in a projective plane PG(2,s2), Mesner (1967) derived a two-class association scheme on the points of the affine space of dimension 3, for which the projective plane is the plane at infinity.We generalize his construction in two ways. We show how his construction works both for nondegenerate and degenerate Hermitian varieties in any dimension.We consider a projective space of dimension N, partitioned into an affine space of dimension N and a hyperplane H of dimension N − 1 at infinity.The points of the hyperplane are next partitioned into 2 or 3 subsets. A pair of points a,b of the affine space is defined to belong to class i if the line ab meets the subset i of H.In the first case, the two subsets of the hyperplane are a nondegenerate Hermitian variety and its complement. In this case, we show that the classification of pairs of affine points defines a family of two-class association schemes. This family of association schemes has the same set of parameters as those derived as restrictions of the Hamming association schemes to two-weight codes defined as linear spans of coordinate vectors of points on a nondegenerate Hermitian variety in a projective space of dimension N − 1. The relations of these codes to orthogonal arrays and difference sets are described in [5,6].In the second case, the three subsets are the singular point of the variety, the regular points of the variety and the complement of the variety defined by a Hermitian form of rank N − 1. This leads to a family of three-class association schemes on the points of the affine space. A geometric construction is first given for the case N = 3.Using a general algebraic method pointed out by the referee, we have also derived the three-class association scheme for general N.