# Zero/positive capacities of two-dimensional runlength-constrained arrays

• Published 2005 in IEEE Transactions on Information Theory

#### Abstract

A binary sequence satisfies a one-dimensional (d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/) runlength constraint if every run of zeros has length at least d/sub 1/ and at most k/sub 1/ and every run of ones has length at least d/sub 2/ and at most k/sub 2/. A two-dimensional binary array is (d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/)-constrained if it satisfies the one-dimensional (d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/) runlength constraint horizontally and the one-dimensional (d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/) runlength constraint vertically. For given d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/,d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/, the two-dimensional capacity is defined as C(d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/) = lim/m,n/spl rarr//spl infin/ log/sub 2/ N(m,n|d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/)/mn where N(m,n|d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/)denotes the number of m/spl times/n binary arrays that are (d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/)-constrained. Such constrained systems may have applications in digital storage applications. We consider the question for which values of d/sub i/ and k/sub i/ is the capacity C(d/sub 1/,k/sub 1/,d/sub 2/,k/sub 2/;d/sub 3/,k/sub 3/,d/sub 4/,k/sub 4/) positive and for which values is the capacity zero. The question is answered for many choices of the d/sub i/ and the k/sub i/.

DOI: 10.1109/TIT.2005.853316

### Cite this paper

@article{Etzion2005ZeropositiveCO, title={Zero/positive capacities of two-dimensional runlength-constrained arrays}, author={Tuvi Etzion and Kenneth G. Paterson}, journal={IEEE Transactions on Information Theory}, year={2005}, volume={51}, pages={3186-3199} }