Noncommutative factorization of variable-length codes

Abstract

We prove a noncommutative version of a theorem of Schiitzenberger on the factorization of variable-length codes. As consequences, we obtain a positive answer to a weak form of the 'factorization conjecture, a complete characterization of maximal and finite codes and a noncommutative extension of an invariance property due to Hansel and Perrin.

On a complete set of operations for factorizing codes

“…Indeed, C is a finite maximal prefix code if and only if C = P (A − 1) + 1 for a finite subset P of A * [3]. In the previous relation, P is the set of the proper prefixes of the words in C. More interesting constructions of factorizing codes can be found in [7] and the result which is closest to a solution of the conjecture, partially reported in Theorem 2.1, was obtained by Reutenauer [5,37,38]. He proved that if we allow that P, S ∈ Z A , then (1) holds for each finite maximal code C.…”

“…Theorem 2.1 [38]. Let C be such that C ∈ N A , (C, 1) = 0 and let P, S be such that P, S ∈ Z A , C = P (A − 1)S + 1.…”

“…The factorization conjecture is one of the most difficult problems which is still open in the theory of codes and the partial results which are known about this conjecture are all reported in Section 2.2. The most impressive result is given by Reutenauer which proves that the above-mentioned equality holds for a finite maximal code C with P, S polynomials with integer coefficients [37,38].…”

On a Property of the Factorizing Codes

“…The first examples of families of factorizing codes can be found in [5,6]. Subsequently, Reutenauer obtained the result that was closest to a solution of the conjecture [4,36,37]. He proved that Eq.…”

A note on the factorization conjecture

“…Only partial results are known (see [2]). The major contribution to this conjecture is due to Reutenauer [26,27]. In particular, he proved that for any finite maximal code C over A, there exist polynomials P, S ∈ Z A such that C −1 = P (A−1)S. We call (P, S) a factorization for C. Moreover we say that a factorization (P, S) for C is positive if P, S or −P, −S have coefficients 0, 1.…”