In this paper, we study the matrix period and competition of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S T {1,…,n−1}, an n×n A=Tn〈S;T〉 is defined to have 1 as (i,j)-entry if only j−i∈S or i−j∈T. We show that maxS+minT≤n minS+maxT≤n, then A has d/d′ where d=gcd(s+t|s∈S,t∈T) d′=gcd(d,minS). Moreover, it shown limit sequence {Am(AT)m}m=1∞ directed sum all ones exce...