  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 from FG to Fn which maps G
to the standard basis of Fn. 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 Fn,
which does not assume an a priori group algebra structure on Fn. 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.
