On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs

Given v objects 1, 2, .. , v, a relation satisfying the following conditions is said to be an association scheme with m classes: (a) Any two objects are either 1st, or 2nd, . . ,or mth associates, the relation of association being symmetrical, i.e., if the object α is the ith associate of the object β, then β is the ith associate of α. (b) Each object a has ni ith associates, the number ni being independent of α. (c) If any two objects α and β are ith associates, then the number of objects which are jth associates of α, and kth associates of β, is pijk and is independent of the pair of ith associates α and β. The numbers v, ni (i = 1, 2, . , m) and pijk (i, j, k = 1, 2, . . . , m) are the parameters of the association scheme. If we have an association scheme with m classes and given parameters, then we get a partially balanced design with r replications and b blocks if we can arrange the v objects into b sets (each set corresponding to a block) such that (i) each set contains k objects (all different) ; (ii) each object is contained in r sets; (iii) if two objects α and β are ith associates, then they occur together in λi sets, the number λi being independent of the particular pair of ith associates α and β.