An analysis on optimal rates of BDB encoders for RLL and MTR systems

In most recording channels, modulation codes are employed to transform user data to sequences that satisfy some desirable constraint. Run-length-limited (RLL(d,k)) and maximum transition run (MTR(j,k)) systems are examples of constraints that improve timing and detection performance. A modulation encoder typically takes the form of a finite-state machine. Alternatively, a look-ahead encoder can be used instead of a finite-state encoder to reduce complexity. Its encoding process involves a delay called look-ahead. If the input labeling of a look-ahead encoder allows block decodability, the encoder is called a bounded-delay-encodable block-decodable (BDB) encoder. These classes of encoders can be viewed as generalizations of the well-known deterministic and block-decodable encoders. Other related classes are finite-anticipation and sliding-block decodable encoders. In this paper, we clarify the relationship among these encoders. We also discuss the characterization of look-ahead and BDB encoders using the concept of path-classes. To minimize encoder complexity, look-ahead is desired to be small. We show that for nonreturn to zero inverted (NRZI) versions of RLL|,(0,k),RLL(1,k), and RLL(d,infin), a BDB encoder does not yield a higher rate than an optimal block-decodable encoder. However, for RLL(d,k) such that dges4 and d+2lesk<infin, we present a BDB encoder with look-ahead one that has a higher rate than any block-decodable encoder. For MTR, we prove that no BDB encoder is asymptotically better than an optimal BDB encoder with look-ahead one