Association Schemes, Orthogonal Arrays and Codes from Non‐Degenerate Quadrics and Hermitian Varieties in Finite Projective Geometries

Abstract

ABSTRACT: In this paper, coexistence of and relations between association schemes, orthogonal arrays and certain families of projective codes have been examined. The projective codes considered are linear spans of a nice projective set P in a hyperplane H = PG ( N -1, s)-such as a quadric or a quadric with its nucleus of polarity or a Hermitian variety. There are two ways to construct association schemes from a projective code. One due to Delsarte (1973) considers the restriction of the Hamming scheme to the code with m weights and if it satisfies Delsarte's condition, an m-class association scheme is obtained by defining two codewords to be i-th associates if the Hamming distance between them is i, i = 0, 1, …, m. The alternative approach, first used by Ray-Chaudburi (1959) and later generalized by Mesner (1967) is to classify points (according to some geometrical criterion) in H = PG(N-1, s) with reference to P. into m types (say). Then, two points of the affine space EG( N, s) (for which H is the hyperplane at infinity) are defined to be i-th associates if the line joining the two points meet H at a point of type i, i = 1, …, m. In many cases, the two association schemes defined with respect to the same projective set have the same parameters. But Examples are given where they do not coincide and, in fact, there are cases where one scheme exists but the other does not.