Runlength Limited Trellis Codes for Partial Response Recording Channels

Partial response models for recording channels go back many years 111. This method of channel modeling has led to an interest in developing trellis codes geared to such channels [2-71. This talk describes a comparison of certain techniques for code construction that combines run-length limiting constraints with constraints that improve the free distance (and thus the noise tolerance) over common models for magnetic recording channels. Three partial res onse channels are considered, "1 El", "1 D and ''1 +D B~ D~~~~ We consider two techniques to define codes and find encoders with run-length constraints and coding gain. In the first constraint type, a convolutional code is used to constrain the locations of the transitions of the signal [4]. In this case, a convolutional code over the ring of integers modulo q, Z, is specified. The code is used to constrain the sequence of transition times, modulo q; the sequence must be a member of the fixed convolutional code. It has been shown how this is an effective technique for finding codes with a non-trivial d constraint (Le., d > 0) 64, 71. The other construction is based on a recent result of Siegel and Karabed, [5-71, and shows more promise in the d=O case. Their result shows that matching the channel null with a null in the codebook leads to a coding gain. Hn this talk we present a comparison of runlength codes we have constructed. The resulting combined constraints are specified by labeled directed graphs. Using Mathematica, we automatically construct a code given a graph specifying a constraint and a rate less than or equal to its capacity. Thus, the main problem is finding interesting constraints. A constraint i s said to have a certain free distance clfree,, for a given partial response channel, if the distance between any two runlength sequences satisfying the constraint have distance at least dfrfe at the 8, output of the channel (and at least one pair of codewords has distance d,,,,). An important issue is the number of states, both in the original constraint and in the final code. The number of states in the constraint determines the complexity of the decoder; we decode these codes by finding the signal satisfying the constraint that is closest to the received signal. (It is possible that a received signal could thus be decoded to something that isn't actually a codeword from the encoder, the probability of making an error of this type is no more than that of picking a codeword that i s incorrect.) The number of states is important for encoding and uncoding. A table is presented that summarizes of the codes that we have found.