An application of Hajós factorizations to variable-length codes

Abstract

We give a new characterization of some factorizations of finite cyclic groups described by Hajós. As a consequence, we prove that the embedding problem for a particular class of codes is decidable. In particular, we link this problem for codes $\text{C ⊆ a}{}^{\mathrm{\ast }}\text{b ∪ ba}{}^{\mathrm{\ast }}$ to Schützenberger's factorization conjecture, via Hajós factorizations.