Ad~z–Disjunct Matrix Designing With Correction Capability Using The Subspace Of The Unitary Space

Geometry of classical Groups over Finite Fields is a kind of very important algebra andgeometry structure. Many scholars structured dz-disjunct matrices with kinds of geometryspace. The mathematical model designed with error detection and correction ability is thedz-disjunct matrix. A d-disjunct matrix is dz-disjunct if a column has at least z 1-entries not covered by the union of any other d columns. A dz-disjunct matrix can detectz-1 errors and correct (z-1)/2 errors.If an extra round of confirmatory tests is allowed, thena dz-disjunct matrix can indeed correct z-1 errors.In this paper,we design a class of newdz-disjunct matrices with the subspaces of the unitary space.In this paper,we structured the binary matrix with the subspaces of the unitary space Fq2n.Let q be a prime power and m,s,r be integers such that 2s≤2m-2≤n+s and m≥r +5≥s + 3. Let Mq2(r,s-4;m,s,n) be the binary matrix whose rows are labeled by the subspacesof type (r,s-4) of Fq2nand columns are labeled by the subspaces of type (m,s) of Fq2n.Mq2(r,s-4;m,s,n) has a 1-entry in row R and column C if and only if R is a subspace of C.In order to discuss the correction capability of the design, we study the following arrangementproblem: For a given subspace S of type(m,s)of unitary space Fq2nand an integer d,we find dsubspaces of type (m-1,s-1)H1,···Hd of S that maximize the number of the subspacesof type (r,s-4) contained in H1∪H2∪···∪Hd.Then with the result ,we give the tighterbound of dz-disjunct matrix.We can get the conclusion:Suppose 2≤d≤qs-1(qs-(-1)s)/(q+1)weconsider the subspaces of type (m-1,s-1)H1,···Hd of S in Fq2n. Let x be the maximalnumber of subspaces of type (m-1,s-1) intersecting in a subspace of type (m-2,s-2)V-S, 2≤x≤d,then|(H1)|ˉ∪···∪(Hd)|ˉ| = dN(r,s-4;m-1,s-1;n) + N(r,s-4;m-2,s-2;n) -dN(r,s-4;m-3,s-3;n) + x(x-d-1)(N(r,s-4; m-2,s-2; n) -N(r,s-4;m-3,s-3;n))With the conclusion, we listed that the Mq2(r,s-4; m; s; n) is not d-disjunct ma-trix when d≥qs-1(qs-(-1)s)/(q+1).Moreover,we discussed the value of z of the dz-disjunct matrixMq2(r,s-4;m,s,n) when 2≤d≤qs-1(qs-(-1)s)/(q+1).