Zero Capacity Region of Multidimensional Run Length Constraints

For integers d and k satisfying 0 d k, a binary sequence is said to satisfy a one-dimensional (d; k) run length constraint if there are never more than k zeros in a row, and if between any two ones there are at least d zeros. For n 1, the n-dimensional (d; k)-constrained capacity is defined as C(n) d;k = lim m1;m2;:::;mn!1 log2N (n;d;k) m1;m2;:::;mn m1m2 mn where N (n;d;k) m1;m2;:::;mn denotes the number of m1 m2 mn n-dimensional binary rectangular patterns that satisfy the one-dimensional (d; k) run length constraint in the direction of every coordinate axis. It is proven for all n 2, d 1, and k > d that C(n) d;k = 0 if and only if k = d + 1. Also, it is proven for every d 0 and k d that limn!1C(n) d;k = 0 if and only if k 2d. This work was supported in part by the National Science Foundation and by a JSPS Fellowship for Young Scientists. A portion of this work was presented in Japanese at the Research Institute for Mathematical Sciences Workshop (RIMS Kokyuroku), Kyoto University, Japan, January 1999. H. Ito is with the Department of Information Science, Faculty of Science, Toho University, Chiba 274-8510, Japan, email: [email protected]. A. Kato is with the Department of Mathematical Engineering and Information Physics, Division of Engineering, University of Tokyo, Tokyo 113-8656, Japan, email: [email protected]. Zs. Nagy and K. Zeger are with the Department of Electrical and Computer Engineering, University of California, San Diego, CA 92103-0407, email: fnagy, [email protected].