Some Results on the Cardinalities of Row Space of Boolean Matrices
ZHONG Li-ping, ZHOU Jian-hui
2008, 23(4):
582-588.
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Let Bn be the set of all n×n Boolean Matrices;R(A) denote the row space of A∈Bn,|R(A)| denote the cardinality of R(A),m,n,k,l,t,i,γi be positive integers,Si,λi be non negative integers.In this paper,we prove the following two results: (1)Let n≥13,n-3≥k > Sl,Si+1> Si,i = 1,2,…,l-1.if k+l≤n,then for any m=2k+2(Sl) + 2(Sl-1)+…+ 2(S1),there exists A∈Bn,such that |R(A)|= m. (2)Let n≥13,n-3≥k>Sn-k-1> Sn-k-2>…>S1>λt>λt-1>…>λ1,2≤t≤n-k.If exist γi(k+1≤γi≤n-1,i=1,2,…,t-1)γi<γi+1 and λt-λt-1≤k-S(n-γ1),λt-i-λt-i-1≤S(n-γi)-S(n-γi+1),i=1,2,…,t-2,then for any m=2k+2(SN-K-1)+2(Sn-k-2)+…+2(S1)+2(λt)+2(λt-1)+…+2(λ1),there exists A∈Bn,as such that |R(A)|=m.
