D-disjunct Matrix. Finite Geometry Construction

In this paper,using a projective plane with n order and an n-dimensional pro-jective space,respectively,we construct a d-disjunct matrix and a(d,e)-disjunctmatrix．Using the vector space over finite fields and 2v-dimensional symplecticspace over finite fields,we construct a(d-m,e)-disjunct matrix and discuss itserror-tolerant decoding and error-correcting capability.It is shown that the error-tolerant decoding and error-correcting capability are increased．The following is our main results．Theorem 3．1．1 Letπbe a projective plane with n order.Suppose M is an²+n+1 matrix,whose rows and columns are indexed by n²+n+1 points,andn²+n+1 lines,respectively．The(i,j)th entry of M is equal to 1 if and only ifthe ith point is contained in the jth line．Then we have1)The matrix M is a n-disjunct matrix with n²+n+1 order；2)The matrix M^T is a n-disjunct matrix with n²+n+1 order.Theorem 3．1．3 Let M be a(0,1)-matrix on the projective planeπwithn order．whose columns are indexed by n²+n+1 lines and rows are indexed byall 2-subsets of n²+n+1 points onπ.M has a 1 in row i and column j if andonly if elements in the ith 2-subsets are on the jth lines．Then1)M is(d,r)-separable matrix；2)M is(d,e)-disjunct matrix,where 1≤d≤n²+n,r=2(?),e=(?)-1．Theorem 3．2．1 Let M be a(0,1)-matrix,whose rows and columns are in- dexed by the set of all l-dimensional subspaces and the set of all m-dimensionalsubspaces,respectively．The(i．j)th entry of M is equal to 1 if and only if the ithl-dimensional subspaces is contained in the jth m-dimensional subspaces．If n>m>l≥1,then M is a(d,e)-disjunct matrix with(?)_q×(?)_q．wherewhere[x]denote the integer at least x．Theorem 4．1．4 Let F_q be the finite field with q elements,where q is a powerof a prime,and let n be a positive integer.Suppose F_q⁽ⁿ⁾ is the n-dimensionalrow vector space over F_q and suppose M is a(0,1)-matrix,whose rows are in-dexed by the set of all d-dimensional subspaces over the F_qⁿ,say A₁,A₂,…A_t,and columns are indexed by the set of all k-dimensional subspaces over the F_qⁿ,say B₁,B₂,…B_n．The(i,j)th entry of M is equal to 1 if and only if A_i(?)B_j．Ifn≥k>d>m≥0,then M is at least(d-m,e)-disjunct,whereTheorem 4．2．2 Let v≥k>d≥1,and let F_q^2v be a 2v-dimensionalsymplectic space．Suppose M is a(0,1)-matrix,whose rows are indexed bythe set of all(d,0)subspaces over the F_q^2v,say A₁,A₂,…A_t,and columns areindexed by the set of all(k,0)subspaces over the F_q^2v,say B₁,B₂,…B_n．The(i,j)th entry of M is equal to 1 if and only if A_i(?)B_j．If v≥k>d>2m≥0,then M is at least(d-m,e)-disjunct,where (?)...