Chapter 6 - Codes

Summary

Introduction

The theory of codes provides some jewels of combinatorics on words that we want to describe in this chapter.

A basic result is the defect theorem (Theorem 6.2.1), which states that if a set X of n words satisfies a nontrivial relation, then these words can be expressed simultaneously as products of at most n — 1 words. It is the starting point of the chapter. In Chapters 9 and 13, other defect properties are studied in different contexts.

A nontrivial relation is simply a finite word w which ambiguously factorizes over X. This means that X is not a code. The defect effect still holds if X is not an ω-code, i.e., if the nontrivial relation is an infinite, instead of a finite, word (Theorem 6.2.4).

The defect theorem implies several well-known properties on words that are recalled in this chapter. For instance, the fact that two words which commute are powers of the same word is a consequence. Another consequence is that a two-element code or more generally an elementary set is an co-code. The latter property appears to be a crucial step in one of the proofs of the DOL equivalence problem.

A remarkable phenomenon appears when, for a finite code X, neither the set X nor its reversal is an ω-code. In this case the defect property is stronger: the n elements of X can be expressed as products of at most n — 2 words (Theorem 6.3.4).