Groups and decompositions of codes

In this paper, some relations between the decompositions of codes and the groups of codes are investigated. We first show the existence of an indecomposable, recognizable, and maximal code X such that the group G(X) is imprimitive, which implies that the answer to a problem put forward by Berstel, Perrin, and Reutenauer in their book “Codes and Automata” is negative. Then, we discuss a special kind of code, that is, rectangular group codes, and show that a completely simple code is a rectangular group code if and only if it can be decomposed as a composition of a complete and synchronized code and a group code.