| In most recording channels, arbitrary user data is encoded into constrained sequences to improve the performance of storage systems. In this work, we focus on three classes of encoders: block, block-decodable, and deterministic. The decoders for block and block-decodable encoders decide the current input symbol based on only the current codeword; thus the error propagation is limited to one block. In contrast, deterministic encoders do not have this property. However, it is much more tractable to compute optimal code rates of deterministic encoders than block and block-decodable encoders. We call these encoders block-type-decodable encoders.; For a constrained system presented by a deterministic graph, the problem of designing a block-type-decodable encoder can be solved by selecting a subset of states of the graph to be used as encoder states. Such a subset is known as a set of principal states. Our goal is to find an optimal set of principal states that yields the highest code rate. We study the relationship between optimal sets of principal states at a finite block length and asymptotically large block length. Specifically, we apply Perron-Frobenius theory to show that for a primitive constrained system and a large enough block length, any optimal set of principal states is also asymptotically optimal. Bounds on block length that guarantee this relationship are given. Characterization of asymptotically optimal sets of principal states is also presented.; An important class of constraints for high-density disk drives is the class of maximum transition run (MTR) systems. Its structure allows us to compute optimal sets of principal states for all block lengths by enumeration.; Finally, we extend our study to the class of bounded-delay-encodable block-decodable (BDB) encoders, which encode by using a look-ahead technique. In some circumstances, this encoder achieves a higher rate than any block-type-decodable encoder and still limits the error propagation to one block. We characterize asymptotically optimal BDB encoders for primitive constraints. For the classes of runlength-limited (RLL) and MTR systems, we investigate how the optimal code rate varies with look-ahead. |
