Constructions Of Association Schemes Based On Singular Linear Spaces And Its Application In Pooling Designs

Association scheme is an important part of algebraic combination,which is closely interrelated with classical groups.Since various geometric spaces under the action of classical groups are well organized and have corresponding counting theorems,association schemes can be constructed easily.Based on the singular linear space over finite fields,an association scheme is constructed,and furtherly Pooling designs and random Pooling designs are given respectively by applying the idea of association schemes.The main work is as follows:First of all,based on singular linear space,we define a set X of subspaces of type(7)m(10)1,1(8)that all contain a fixed subspace of type(7)m-1,0(8).According to the different types of the sum of any two subspaces in X,an association scheme with a class number of 5 on X is constructed.And all parameters of the scheme are computed by means of combinatorial counting method.Secondly,taking the singular linear space as a theoretical tool,a family of Pooling designs is given based on the idea of constructing association schemes,and its disjunctive properties are obtained.In addition,by analyzing its fault-tolerant and error-correction performance,it concludes that the new designs is superior to the existing ones under the same test efficiency.Last,two types of random Pooling designs with error tolerance are constructed based on the related theory of singular linear space and the construction of the above Pooling designs.By studying their disjunct properties,all parameters of the random Pooling designs are obtained.