The Association Scheme Based On Quasi-orthogonal Groups And Its Application

Let Fq be the finite field with q elements,and m be a positive integer,#12 be the m × m matrix over Fq,where Im is the identity matrix of m order,#12 Obviously,under the matrix multiplication operation,all m × m matrices P satisfying P?P'=? on Fq form a group,called quasi-orthogonal groups and denoted QO(n,q).When q and m are both even,let A1,A2 be m × m circulant matrices over Fq,set S={(A1,A2)|A2KmA'1+A1A'2=0},where A'1 is the transpose of A1,then the group QO(n,q)acts transitively on the set S.Furthermore,the corresponding associaton scheme(?)n,q=(S,{?i}0?i?d)is determined.In this paper,when q is even we determine the structure of the quasi-orthogonal group QO(n,q),determine the associaton classes of associaton scheme(?)n,q,and calculate the intersection number of(?)8,2.Using the set S to construct a class of regular digraph Fn,q,we discuss the relationship between graphs ?n,q and weakly distance-regular digraph.