Periodic-finite-type shift spaces

Abstract

We introduce the class of periodic-finite- type (PFT) shift spaces. They are the subclass of shift spaces defined by a finite set of periodically for- bidden words. Examples of PFT shifts arise naturally in the context of distance-enhancing codes for partial- response channels. We show that the set of PFT shift spaces properly contains the set of finite-type shift spaces and is properly contained in the set of almost- finite-type shift spaces. We prove several properties of labeled graphs that present PFT shifts. For a given PFT shift space, we identify a finite set of forbidden words - referred to as periodic first offenders - that define the shift space and that satisfy certain mini- mality properties. Finally, we present an efficient al- gorithm for constructing labeled graphs that present PFT shift spaces. I. INTRODUCTION Magnetic recording systems often make use of binary codes that disallow the appearance of certain sequences that are problematic in the data recording or retrieval process. Recent disk drives have incorporated distance-enhancing constrained codes that forbid the appearance of certain patterns in a pe- riodic manner. In this paper, we examine properties of such time-varying constraints, and establish their relationship to more familiar classes of constrained systems. Specifically, we consider bi-infinite sequences of sym-