Constructing D~z-disjunct Matrices With The Subspaces Of Unitary Space And Symplectic Space

The error-correcting pooling design is widely used in coding theory,computer network,blood detection and the study of molecular biology.In this paper,two new types of dz-disjunct matrices are constructed respectively on unitary space and symplectic space,so as to facilitate the practical applications.Firstly,this paper introduces the basic theory,basic mathematical model,development and research status of group testing,and then introduces the definitions and some counting theorems of unitary space and symplectic space.Thus,the ideas about constructing dz-disjunct matrices on unitary space and symplectic space are obtained.In chapter 2,a new class of dz-disjunct matrices are constructed by using the inclusion relations between the subspace of type(r,s-2)and the subspace of type(m,s)in unitary space over finite fields,where q is the power of an odd prime,m,s,r are integers satisfying m>r+3>s+5,s>2,and s is odd.The rows of such dz-disjunct matrices are marked by the subspaces of type(r,s-2)and the columns are marked by the subspaces of type(m,s).The element of dz-disjunct matrices in row R and column C is 1 when R is a subspace of C,otherwise it is 0.Firstly,the scope of discussion is reduced by comparing the number of the subspace of type(r,s-2)contained in any subspace of type(m-1,s),any subspace of type(m-1,s-1)and any subspace of type(m-1,s-2).Then,the counting theorems in unitary space are use to discuss and determine the conditions met by parameters m,s,r to make Mq2(r,s-2;m,s;n)become dz-disjunct matrices,and the value range of d as well as the lower bound of z under these conditions.In addition,the upper and lower bounds of z are given for a special case.In chapter 3,a new class of dz-disjunct matrices are constructed by using the inclusion relations between two different types of subspaces in symplectic spaces over finite fields,where d is an integer satisfying d? 2.m,s and r are integers satisfying m? r+6? 9,m?2s+1? 5,r? s-2.The row and column of this kind of dz-disjunct matrices are marked respectively by the subspace of type(r,s-2)and the sub space of type(m,s)in symplectic space.Moreover,every matrix of this kind is a 0-1 matrix,and the element at the intersection of rows and columns is 1 if the sub space identifying the row is included in the subspace identifying the column,otherwise it is 0.In this paper,we first transform the problem to reduce the scope of discussion.Then we use the counting theorems in symplectic space to calculate and compare to further narrow the scope of discussion.After that,we use the inclusion-exclusion principle and the counting theorems in symplectic space to determine the value range of parameter d and the lower bound of z.In addition,the value ranges of z are given for a special case.