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.

翻译：长度为n的（左）群码是一种线性码，它是群代数中（左）理想的像，通过从FG到Fn的同构将G映射到Fn的标准基。许多经典线性码已被证明是群码。本文基于Fn空间的内在属性，建立了一个判定线性码是否为群码的准则，该准则无需预先假定Fn上具有群代数结构。作为应用，我们给出了一类群（包括亚循环群），其中每个双边群码都是阿贝尔群码。众所周知，Reed-Solomon码是循环码，其奇偶校验扩展是初等阿贝尔群码。这两类码均包含在Cauchy码类中。利用我们的准则，我们对某些长度的左群码Cauchy码进行分类，并确定了这些码上可能的群码结构。