An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism $${\mathbb{F}G \rightarrow \mathbb{F}^n}$$ which maps G to the standard basis of $${\mathbb{F}^n}$$ . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space $${\mathbb{F}^n}$$ , which does not assume an “a priori” group algebra structure on $${\mathbb{F}^n}$$ . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.