Some Results on the Cardinalities of Row Space of Boolean Matrices

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＋〉Si,i=1,2…,l-1.if k＋l≤n,then for any m=2^k＋2^S1-l＋…＋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-Sn-γ1,λt-i-λt-i-1≤Sn-γi-Sn-γii＋1,i=1,2…,t-2,then for any m=2^k＋2^Sn-k-1＋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.