Partial characterization of the positive capacity region of two-dimensional asymmetric run length constrained channels

A binary sequence satisfies a one-dimensional (d,k) run length constraint if every run of zeros has length at least d and at most k. A two-dimensional binary pattern is (d₁,k₁,d₂ ,k₂)-constrained if it satisfies the one-dimensional (d ₁,k₁) run length constraint horizontally and the one-dimensional (d₂,k₂) run length constraint vertically. For given d₁, k₁, d₂, and k ₂, the asymmetric two-dimensional capacity is defined as C _d1,k1,d2,k2=lim_m,n→∞ (1/(mn)) log₂ N_m,n^{(d1,k1,d2,k2)} where N_m,n^{(d1,k1,d2,k2)} denotes the number of (d₁ ,k₁,d₂,k₂)-constrained m×n binary patterns. We determine whether the capacity is positive or is zero, for many choices of (d₁,k₁,d₂,k ₂)