利用酉空间中的子空间构造d^z-析取矩阵

Constructing d^z-disjunct matrices with subspaces of unitary space

容错的Pooling设计数学模型是d^z-析取矩阵,在实践中具有广泛的应用。在有限域上的酉空间中,利用2种不同类型的子空间之间的包含关系构建了一类d^z-析取矩阵。这类d^z-析取矩阵的行和列分别由这2种不同类型的子空间来标识,而且该类矩阵是0-1矩阵,如果标识该行的子空间包含在标识该列的子空间当中,其行列交叉处的元素为1,否则为0。首先,将问题进行了转化,从而缩小了讨论范围;接着,利用酉空间中的计数定理进行计算和比较,进一步缩小了讨论范围;之后,利用容斥原理以及酉空间中计数定理确定了d^z-析取矩阵中参数d的取值范围;最后,利用此结果确定了反映d^z-析取矩阵纠错能力的参数z,即证明了z的存在性及紧性。

The mathematical model of an error-tolerant pooling design is a d^z-disjunct matrix,which is widely used in practice.A class of d^z-disjunct matrices are constructed by using the inclusion relation between two different types of subspaces in unitary space over a finite field.The row and column of this class of d^z-disjunct matrices are indexed respectively by the two different types of subspaces,and the matrices of this class are 0-1 matrices,their element at the intersection of the row and column is 1 if the subspace indexing the row is included in the subspace indexing the column,otherwise it is 0.Firstly,the problem is transformed to narrow the scope of the discussion,and then the counting theorem in unitary space is used for calculation and comparison to further narrow the scope of the discussion.After that,the value range of the parameter d in the d^z-disjunct matrices is determined by using the inclusion-exclusion principle and the counting theorem in unitary space.Finally,the parameter z which reflects the error correction capability of the d^z-disjunctmatrices is determined by using this result,that is,the existence and compactness of z are proved.