Finite Ring Z <sub> Pq And </ Sub> On A Staggered Matrix Combination Program

Association schemes have close connections with coding theory ,design theory and finite group theory, etc. At present, they have become important and interesting objects in their own right. We refer the reader to [9] for the general theory on association schemes.As early as in 1965, Wan Zhe-xian constructed a class of association schemes using all n x n Hermitian matrices over a finite field Fq2 by setting (A, B) Ri iff rank(A-B)=i for any two nxn Hermitian matrices A and B, where n is a given natural number. He also computed the parameters of this association scheme in the case n = 2. Wang Yang-xian (1980) computed the parameters of Wan's association scheme in the general case and generalized the methods of comstructing an association scheme to the cases of alternate matrices and of general m x n matrices over a finite field. Later, Huo Yuanji and Zhu Xueli (1987), Huo Yuanji and Wan Zhe-xian (1993) Wang Yang-xian and Ma Jianmin succeedingly discussed the association schemes based on the symmetric matrices over a finite field of odd or even characteristic. In this paper, we discuss the association schemes based on the alternate matrices over the finite ring Zpq , where p, q are two different primes, and their parameters are computed.Let Zpq denote the residue class ring mod pq , where p, q are two different primes, any element of Zpq is denoted by a, and p < q,ps < q < ps+1, where s is some positive integer. Denote the multiplicative group of Zpq by Zpq* , which is constructed by all the inverse elements of Zpq under the multiplication. An nxn matrix is called alternate , if aij + aji = 0, for i j, and an = 0(1 < i, j < n). Denote the set of all n x n matrices overZpq by (Zpq)n and the set of all n x n inverse matrices over Zpq by GLn(Zpq), respectively. And let Xn denote the set of all n x n alternate matrices over Zpq. Let T0 denote the traslation group of Xn that acts on Xn in the following way:Let G be the direct product of GLn(Zpq) and T0, G acts on Xn transitively in the following way:then (G, Xn) determines an association scheme For X, Y Xn,we setting: (X,Y) R(rl,rp)(R ) if and only if X - Y is congruent toFinally we can concluded that is a symmetric association scheme containing [n/2]2 + 2[n/2] association classes, and its parameters are calculated by using the definitions of direct product and isomophism of association schemes.