Association Schemes Based On Singular Linear Space And Singular Symplectic Space And Its Applications

Association scheme appeared with partial balance incomplete block designs is a combination structure. It depicts some balance among different users who possess several association relations. Since there are close relationships among association schemes, code theory, graph theory and finite field, the study of association schemes has been an important part of algebraic combination. Moreover, geometry of classical groups over finite fields combined with good combination structure counting ways,which made it easily to construct association relations. In this paper, the construction of association schemes based on singular linear spaces and the subspaces in singular symplectic space are provided in sequence. Using the matrix calculus methods and combinatorial design methods, all parameters of the association scheme are computed.Furthermore, an authentication code with perfect secrecy can be derived from the constructed association scheme.A new symmetric association schemes based on the subspaces of type(m ?s,s)in singular linear spaces is constructed in the article, all parameters of the association scheme are computed by using the matrix calculus methods and combinatorial design methods. Moreover,we obtain an authentication code with perfect secrecy property from the association scheme.Another new symmetric association schemes based on the subspaces of type?2,0,1? in singular symplectic space of class 7 is constructed in the article ? All parameters of the association scheme are computed by using the matrix calculus methods.