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Entropy of a bit-shift channel

We consider a simple transformation (coding) of an iid source called a bit-shift channel. This simple transformation occurs naturally in magnetic or optical data storage. The resulting process is not Markov of any order. We discuss methods of computing the entropy of the transformed process, and study some of its properties. Results presented in this paper originate from the discussions we had at the "Coding Club" - the weekly seminar on coding theory at the Philips Research Laboratories in Eindhoven. Mike Keane, when his active travelling schedule permits, is also attending this seminar. We would like to use this opportunity to thank Mike for his active participation, pleasant and fruitful discussions, his inspiration which we had a pleasure to share. 1. Bit-shift channel In this paper we consider a simplified model for errors occurring in the readout of digital information stored on an optical recording medium like the Compact Disk (CD) or the Digital Versatile Disk (DVD). For more detailed information on optical storage see (9) or (16). On optical disks the information is stored in a reflectivity pattern. For technical reasons, it is advantageous to use only two states, i.e. "low" and "high" reflectivity. Figure 1 shows the disk surfaces for two types of the DVD's. While the presence of only 2 states greatly simplifies the detection of the state, it reduces the maximum spatial frequency, and hence storage capacity. In this situation it is better not to encode the information in the reflectivity state itself but rather in the location of the transitions: The reflectivity pattern consists of an alternating sequence of "high" and "low" marks of varying length (an integer multiple of some small length unit), while each mark exceeds a minimal length, say d + 1 units. Hence, this "run-length limited" (RLL) encoding makes sure no mark is too short for the disk while the information density is only limited by the accuracy of determining the length of the marks, or equivalently the location of the transitions. For technical reasons (to recover the length unit from the signal itself)