Decomposition and construction of group codes

A decomposition of a group code into a normal subcode and a quotient code can be characterized as group extension (extension code) of the subcode and the quotient code. Simple necessary and sufficient conditions are given for the existence of an extension code with given sub- and quotient code. These purely algebraic results are complemented by a characterization of finitely generated, complete, k-controllable group codes by a sublength-generation property of finite-length generators. This property leads to a new construction process of canonical minimal encoders