Two-dimensional runlength constrained arrays with equal horizontal and vertical constraints

A binary array is a (d₁,k₁,d₂,k ₂) runlength constrained array if the runs of zeros in every row and column have length at least d₁ and at most k₁ , and the runs of ones in every row and column have length at least d₂ and at most k₂. Such arrays arise in the context of digital storage devices. Writing N(m,n|d₁,k₁ ,d₂,k₂) for the number of (d₁,k ₁,d₂,k₂) runlength constrained arrays of size mtimesn, the capacity C(d₁,k₁,d₂ ,k₂) is defined to be lim_m,nrarrinfin(1/mn)log₂N(m,n|d₁,k ₁,d₂,k₂). Let d₂ be an integer such that d₂ ges 1. The paper shows that C(1,d₂+delta,d₂,d₂) is positive when delta ges 2, but is zero when delta les 1