"On Linear Associative Algebras Corresponding to Association Schemes of" by R. C. Bose and Dale M. Mesner

Mathematics, Department of

 

Document Type

Article

Date of this Version

March 1959

Comments

Published in The Annals of Mathematical Statistics, Vol. 30, No. 1. (Mar., 1959), pp. 21-38. Copyright © 1959 Institute of Mathematical Statistics. Used by permission. The Annals of Mathematical Statistics is superceded by The Annals of Statistics, online at http://www.imstat.org/aos/

Abstract

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.

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