Capacity of M-ary 2-D RLL constraints for optical recording channels

A new class of M-ary two-dimensional (2-D) run-length limited (RLL) constraints is introduced in this paper. In particular, only RLL d constraints on 2D m × n arrays with limited height m are considered. Column vectors of 2-D arrays are used as states in a directed graph. The transfer matrix of the graph is obtained with consideration of non-binary alphabet, and the largest eigenvalue of the transfer matrix is used to calculate the capacity of 2-D (M, d, ∞) RLL constraints on 2-D arrays. Some numerical results are derived for RLL d = 1 and d = 2 constraints. We also present an example of efficient finite-state encoder for d = 1 constraint by state splitting and merging algorithm. The 2-D (M, d, ∞) constrained codes have potential to be applied in advanced 2-D optical recording channels.