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 (d1; k1) run length constraint horizontally and the one-dimensional (d2; k2) run length constraint vertically. For given d1; k1; d2, and k2, the asymmetric two-dimensional capacity is defined as Cd1;k1;d2;k2 = limm;n!1 (1=(mn)) log2N (d1;k1;d2;k2) m;n where N (d1;k1;d2;k2) m;n 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; k2). This work was supported in part by the National Science Foundation and by the Japan Society for the Promotion of Science. A. Kato is with the Department of Mathematical Engineering and Information Physics, Graduate School of Engineering, University of Tokyo, Tokyo 113-8656, Japan, ([email protected]). K. Zeger is with the Department of Electrical and Computer Engineering, University of California, San Diego, CA 92093-0407, ([email protected]).