Capacity Bounds for the 3-dimensional (0,1) Runlength Limited Channel

The capacity C (3) (0;1) of a 3-dimensional (0; 1) runlength constrained channel is shown to satisfy 0:522501741838 C (3) (0;1) 0:526880847825 . The authors are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla CA 92093-0407. Email: fnagy,zegerg@code.ucsd.edu . The research was supported in part by the National Science Foundation. 1 Introduction A binary sequence satisfies a 1-dimensional (d; k) runlength constraint if there are at most k zeros in a row, and between every two consecutive ones there are at least d zeros. An n-dimensional binary array is said to satisfy a (d; k) runlength constraint, if it satisfies the 1-dimensional (d; k) runlength constraint along every direction parallel to a coordinate axis. Such an array is called valid. The number of valid n-dimensional arrays of size m 1 \Theta m 2 \Theta : : : \Theta m n is denoted by N (d;k) m1;m2 ;:::;mn and the corresponding capacity is defined as C (n)..