酉空间上一类析取矩阵的构造及紧界分析
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摘要
析取矩阵主要用来检测样本空间中的阳性样本,也称为问题样本,而且每个析取矩阵都是一个(0,1)矩阵。目前有许多文献利用有限域上的几何空间(简称有限几何空间)来构作d~z-析取矩阵,其中辛空间中的结果较多。在这些文献中,有一些是利用有限几何空间中的子空间之间的包含关系来构作d~z-析取矩阵的,并且讨论了试验效率(d~z-析取矩阵的行数与列数之比)及z的紧界。用酉空间F■的(m,s)-型子空间标识d~z-析取矩阵的行,(r,s-1)-型子空间标识d~z-析取矩阵的列,利用它们之间的包含关系构作了一类新的d~z-析取矩阵。通过求包含在一个(m,s)-型子空间中的、d个(m-1,s-1)-型子空间里的、(r,s-1)-型子空间个数的最大值,给出了d和z的取值范围及z的紧界。由于(r,s-1)-型子空间中的s-1与(m,s)-型子空间中的s相差较小,所以本文能够相对较快地得到了d和z的的取值范围及z的紧界
Disjunction matrices mainly are used to test positive samples in sample space, which are also known as the problem samples, and every disjunction matrix is a(0,1) matrix. At present, many literatures make use of geometric spaces over a finite field to construct d~z-disjunction matrices, in which there are more results in Symplectic Space. Some of these literatures make use of the inclusion relation between subspaces in geometric spaces over a finite field to construct d~z-disjunction matrices, the test ratio(the ratio of the rows to columns of the d~z-disjunction matrices) and the tighter bound of z are discussed. This paper uses the subspaces of the type of(m,s) in Unitary Space F■ to index the rows of the d~z-disjunction matrices, the subspaces of the type of(r,s-1) in Unitary Space F■ to index the columns of the d~z-disjunction matrices, using the inclusion relation between them to construct a new class of d~z-disjunction matrices. By means of calculating the largest number of the subspaces of the type of(r,s-1) included in d subspaces of the type of(m-1,s-1), which included in a given subspace of the type of(m,s), we give the value range of d and z and the tighter bound of z. Since the gap between s-1 in the subspaces of the type of(r,s-1) and s in the subspaces of the type of(m,s) is smaller, the value range of d and z and the tighter bound of z can be relatively quick obtained in this paper.
Disjunction matrices mainly are used to test positive samples in sample space, which are also known as the problem samples, and every disjunction matrix is a(0,1) matrix. At present, many literatures make use of geometric spaces over a finite field to construct d~z-disjunction matrices, in which there are more results in Symplectic Space. Some of these literatures make use of the inclusion relation between subspaces in geometric spaces over a finite field to construct d~z-disjunction matrices, the test ratio(the ratio of the rows to columns of the d~z-disjunction matrices) and the tighter bound of z are discussed. This paper uses the subspaces of the type of(m,s) in Unitary Space F■ to index the rows of the d~z-disjunction matrices, the subspaces of the type of(r,s-1) in Unitary Space F■ to index the columns of the d~z-disjunction matrices, using the inclusion relation between them to construct a new class of d~z-disjunction matrices. By means of calculating the largest number of the subspaces of the type of(r,s-1) included in d subspaces of the type of(m-1,s-1), which included in a given subspace of the type of(m,s), we give the value range of d and z and the tighter bound of z. Since the gap between s-1 in the subspaces of the type of(r,s-1) and s in the subspaces of the type of(m,s) is smaller, the value range of d and z and the tighter bound of z can be relatively quick obtained in this paper.
引文
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[ 2 ]GUO J,WANG K S.Pooling designs with surprisingly high degree of error correction in a finite vector space[J].Discrete Applied Mathematics,2012,160:2172-2176.
[ 3 ]LIU X M,GAO X.New error-correcting pooling designs with vector spaces over finite field[J].Discrete Mathematics,2015,338:857-862.
[ 4 ]ZHANG G S,SUN X L,LI B L.Error-correcting pooling designs associated with the dual space of unitary space and ratio efficiency comparison[J].Journal of Combinatorial Optimization,2009,18:51-63.
[ 5 ]GUO J.Pooling designs associated with unitary space and ratio efficiency comparison[J].Journal of Combinatorial Optimization,2010,19:492-500.
[ 6 ]GAO S G,LI Z T,YU J C,et al.DNA library screening,pooling design and unitary spaces[J].Theoretical Computer Science,2011,412:217-224.
[ 7 ]张更生,曹晓艳.酉空间中一类子空间的排列问题与具有纠错能力的dz-析取矩阵[J].数学学报中文版,2012,55(3):385-388.
[ 8 ]高星.基于奇异线性空间和奇异酉空间的Pooling设计的构造[D].天津:中国民航大学,2015:9-27.
[ 9 ]WAN Z X.Geometry of classical group over finite fields[M].Beijing:Science Press,2002.
[10]赵向会,李莉,张更生.辛空间的排列问题及具有容错能力的Pooling设计的紧界[J].数学物理学报,2012,32A(2):414-423.
[11]ZHAO X H,XU N,ZHANG G S.New pooling design constructed with pseudo-symplectic space[J].Chinese Quarterly Journal of Mathematics,2012,27(4):526-534.
[12]赵向会,刘铭,张更生.利用伪辛子空间构造的一类Pooling设计及其纠错能力的紧界分析[J].河北师范大学学报(自然科学版),2015,39(5):369-373.
[13]张晓娟.基于辛几何和伪辛几何的子空间构造压缩感知矩阵[D].天津:中国民航大学,2015.
[14]赵燕冰,刘志刚.奇特征正交空间上可检错Pooling设计的构作[J].湖南理工学院学报(自然科学版),2009,22(3):16-18.
[15]郭秀平.正交空间上子空间的排列问题和可纠错Pooling设计的讨论[D].石家庄:河北师范大学,2009:8-32.