In this paper, we analyze the distance properties of two-dimensional (2-D) intersymbol interference (ISI) channels, in particular the 2-D partial response class-1 (PR1) channel which is an extension of the one-dimensional (1-D) PR1 channel. The minimum squared-Euclidean distance of this channel is proved to be 4 and a complete characterization of the squared-Euclidean distance 4 error events is provided. As for 1-D channels, we can construct error-state diagrams for 2-D channels to help characterize error events. We propose an efficient error event search algorithm operating on the error-state diagram that is applicable to any 2-D channel.

A two-dimensional (2-D) constrained system consists of 2-D binary sequences satisfying certain properties. Most of the 2-D constrained systems can be represented as a set of forbidden patterns, which should be avoided in the input sequences to the channel. This paper proposes a depth-first algorithm to search for single-state codes with rectangular codewords. The algorithm provides the optimal single-state codes, meaning that no other single-state code can have larger coding rate. For certain 2-D constraints, this algorithm has comparatively low complexity. The coding rate asymptotically approaches the capacity of the 2-D constrained system when the codeword size increases.

For one-dimensional (1-D) recording channels, the detector performance can be assessed by the distance properties of the channel, which are defined in terms of the maximum-likelihood decoding trellis. Error events with minimum and near minimum distances play an important role in the performance of the recording system particularly at moderate to high signal-to-noise ratio (SNR). In this paper, we analyze the distance properties of two-dimensional (2-D) inter-symbol interference channels, in particular the 2-D PR1 channel which is an extension of the 1-D PR1 channel. The minimum distance of this channel is proved to be 2 and a complete characterization of the distance-2 error events is provided. Also, the error events with squared-Euclidean distance 6 are partially characterized. Analogous to 1-D channels, error-state diagrams for 2-D channels can be constructed to characterize the error events. We propose an efficient error event search algorithm operating on the error-state diagram that is applicable to any 2-D channel

The hard-square model is a two-dimensional (2-D) constrained system consisting of all binary arrays on a rectangular grid in which 1's are isolated both horizontally and vertically. This paper proposes algorithms to search for single-state and finite-state block codes with rectangular codewords that satisfy the hard-square constraint. Although the codeword size is small, single-state block codes with coding rates larger than 0.5 can be designed. Letting the 2-D constrained sequences have finite memory increases the achievable coding rate. A method for designing low-complexity encoders and decoders is also presented. When the codeword size increases, the coding rate asymptotically approaches the capacity but with rapidly increasing complexity of the encoder and decoder.

The hard-square model is a two-dimensional (2D) constrained system consisting of all binary 2D rectangles in which 1's are isolated both horizontally and vertically. We propose 2D block codes with rectangular codewords which achieve coding rates larger than 0.5.