# Optimal block-type-decodable encoders for constrained systems

## Abstract

A constrained system is presented by a finite-state labeled graph. For such systems, we focus on block-type-decodable encoders, comprising three classes known as block, block-decodable and deterministic encoders. Franaszek gives a sufficient condition which guarantees the equality of the optimal rates of block-decodable and deterministic encoders for the same block length- In this paper, we introduce another sufficient condition, called the straight-line condition, which yields the same result, Run-length limited RLL(d, k) and maximum transition run MTR(j, k) constraints are shown to satisfy both conditions, In general, block-type-decodable encoders are constructed by choosing a subset of states of the graph to be used as encoder states. Such a subset is known as a set of principal states. For each type of encoder and each block length, a natural problem is to find a set of principal states which maximizes the code rate. We show how to compute the asymptotically optimal sets of principal states for deterministic encoders and how they are related to the case of large but finite block lengths. We give optimal sets of principal states for MTR(j, k)-block-type-decodable encoders for all codeword lengths. Finally we compare the code rate of nonreturn to zero inverted (NRZI) encoders to that of corresponding nonreturn to zero (NRZ) and signed NRZI encoders.