Zero/Positive Capacities of Two-Dimensional Runlength-Constrained Arrays

A binary sequence satisfies a one-dimensional runlength constraint if every run of zeros has length at least and at most and every run of ones has length at least and at most . A two-dimensional binary array is -constrained if it satisfies the one-dimensional runlength constraint horizontally and the one-dimensional runlength constraint vertically. For given , the two-dimensional capacity is defined as $$displaylines C(d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4) hfillcr hfill=, lim_m,n rightarrow infty log_2 N(m, n ,vert, d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)over mn $$ where $$N(m, n ,vert, d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)$$ denotes the number of binary arrays that are -constrained. Such constrained systems may have applications in digital storage applications. We consider the question for which values of and is the capacity positive and for which values is the capacity zero. The question is answered for many choices of the and the .