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 FG→ F which maps G to the standard basis of F. 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 F, which does not assume an “a priori” group algebra structure on F. 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. In this paper F = Fq denotes the field with q elements where q is a power of a prime p. We consider F as the alphabet of linear codes of length n and so the ambient space is Fn, the n-dimensional vector space. The standard basis of Fn is denoted by E = {e1, . . . , en}. For any group G, we denote by FG the group algebra over G with coefficients in F. Recall that a linear code C ⊆ Fn is said to be cyclic if and only if C is closed under cyclic permutations, that is, (x1, . . . , xn) ∈ C implies (x2, . . . , xn, x1) ∈ C. For Cn = 〈g〉, the cyclic group of order n, the bijection π : E → Cn given by π(ei) = gi−1 extends to an isomorphism of vector spaces φ : Fn → FCn and the cyclic codes in Fn are the subsets C of Fn such that φ(C) is an ideal of FCn. If G is a group of order n and C ⊆ Fn is a linear code then we say that C is a left G-code (respectively, a right G-code; a G-code) if there is a bijection φ : E → G such that the linear extension of φ to an isomorphism φ : Fn → FG maps C to a left ideal (respectively, a right ideal; a two-sided ideal) of FG. A left group code (respectively, group code) is a linear code which is a left G-code (respectively, a G-code) for some group G. A (left) cyclic group code (respectively, abelian group code, solvable group code, etc.) is a linear code which is (left) G-code for some cyclic group (respectively, abelian group, solvable group, etc.). In general, if G is a class of groups then we say that a linear code C is a (left) G group code if and only if C is a (left) G-code for some G in G. Note that the cyclic codes of length n are Cn-codes. However, not every Cn-code is a cyclic code. For example, the linear code {(a, a, b, b) : a, b ∈ F} is not a cyclic code but it is a C4-code via the map φ : {e1, . . . , e4} → C4 given by φ(e1) = 1, φ(e2) = g2, φ(e3) = g and φ(e4) = g3. If one fixes a bijection φ : E → G to induce an isomorphism φ : Fn → FG then the (left) G-codes are precisely those codes of Fn which are permutation equivalent to codes of the form φ−1(I),