An Arrangement Problem Of Subspaces Of Orthogonal Space And Discussion Of An Error-detecting Pooling Design

The mathemafical model of pooling designed with error detection and correction ability is the dz-disjunct matrix.Many scholars structured dz-disjunct matrices with some geometry spaces on the geometry of classical groups over finite fields.In this paper,we design a class of new dz-disjunct matrices with the subspaces of the orthogonal space Fq(2v+δ).Firstly,we structured the binaty matrix with the subspaces of the orthogonal space Fq(2v+δ) Let q be a prime power and m,s,r be integers such that v≥m≥r+7≥2s+4,let Mq(r,2(s-3),s-3；m,2s,s；2v+δ,△)be the binary matrix whose rows are indexed by subspaces of type(r,2(s-3),s-3)of Fq(2v+δ) and columns are indexed by subspaces of type (m,2s,s)of Fq(2v+δ).Obviously,the binary matrix is a dz-disjunct matrix.Secondly,in order to discuss the correction capability of the design,we study the following arrangement problem:For a given subspace Co of type(m,2s,s)of orthogonal space Fq(2v and an integer d(2≤d≤q2+q+1),we find d subspaces of type(m-1,2(s-1),s-1)K1,…Kd of C0that maximize the number of the subspaces of type(r,2(s-3),s-3) contained in K1(?) K2(?)…(?) Kd.Finally,on the base of these prepared work we gain the principal conclusion of this paper: Suppose d≤(qs-1)(qs-1-1)/q-1,we consider the subspaces of type(m-1,2(s-1),s-1)K1,…Kd of C0in Fq(2v+δ).Let x be the maximal number of subspaces of type(m-1,2(s-1),s-1) intersecting in a subspace of type(m-2,2(s-2),s-2)V (?) C0,2≤x≤d,then|K1(?)…(?)Kd|≤dN(r,2(s-3),s-3；m-1,2(s-1),s-1；2v+δ,△)+N(r,2(s-3),s-3;m-2,2(s-2),s-2;2v+δ,△)-dN(r,2(s-3),s-3;m-3,2(s-3),s-3:2v+δ,△)+x(x-d-1)[N(r,2(s-3),s-3；m-2,2(s-2),s-2；2v+δ,△)-N(r,2(s-3),s-3;m-3,2(s-3),s-3;2v+δ,△)]. With the conclusion,welisted that the Mq(r,2(s-3),s-3；m,2s,s；2v+δ,△)is not d-disjunct matrix when d≥(qs-1)(qs-1-1)/q-1.Moreover,we discussed the value of z of the dz-disjunct matrix Mq(r,2(s-3),s-3；m,2s,s；2v+δ,△)when2≤d≤(qs-1)(qs-1-1)/q-1.