Algorithms for sliding block codes - An application of symbolic dynamics to information theory

Abstract

Abstract--Ideas which have origins in Shannon's work in information theory have arisen independently in a mathematical discipline called symbolic dynamics. These ideas have been refined and developed in recent years to a point where they yield general algorithms for constructing practical coding schemes with engineering applications. In this work we prove an extension of a coding theorem of Marcus and trace a line of mathematics from abstract topological dynamics to concrete logic network diagrams.

Chains, entropy, coding

“…The almost sofic systems may be considered to be among the most useful ones for the purposes of encoding data. Frequently the sequences which result from coding have to satisfy certain restrictions, for example to make them suitable for magnetic recording (see [2] and [17].) These restrictions might be summarized by requiring the image of the coding to be an SFT, a sofic system, or another subshift.…”

“…Here one way to approximate X from the inside in entropy by SFT's is with the 2-shift {a l , a 2 ) z . Another way is with the SFT's 2 m consisting of all sequences whose sub-blocks are sub-blocks of concatenations of the blocks {a t , a 2 }"b", 1 < n < m.…”

“…The fc'th returning (leftward) path consists of r k edges which are all labelled 0. Then this is a connected graph whose labelling X contains the 2-shift 2 2 a n d hence equals 1 2 •…”

“…Of course one seeks 424 K. Petersen codes that are efficient, have limited error propagation, and can be easily implemented-or, more realistically, ones with a compromise among these competing attributes. See [2], [17], [11], [12], [16], [21], [26], [19] for some of the previous work on this sort of problem. A further restriction arises from attempts to control the power spectrum of the encoded signal.…”

“…All of these systems are intrinsically ergodic -they have unique (shift-invariant Borel probability) measures of entropy equal to the topological entropy. For a survey of the useful properties of these kinds of systems as well as an explanation of how they come up in problems of magnetic recording, see [2] and [17].…”