Group-algebra code[1] 

An \( [n,k]_q \) code based on a finite group \( G \) of size \(n \). A group-algebra code for an Abelian group is called an Abelian group-algebra code.

Group algebra

For a given field \(\mathbb{F}_q\) and a finite group \(G\) of order \(|G|=\ell\), the group algebra (a ring) \(\mathbb{F}_q[G]\) is defined as an \(\mathbb{F}_q\)-linear space of all formal sums \begin{align} \label{eq:algebra-element} x\equiv \sum_{g\in G}x_g g,\quad x_g\in \mathbb{F}_q, \tag*{(1)}\end{align} where group elements \(g\in G\) serve as basis vectors, equipped with the product naturally associated with the group operation, \begin{align} \label{eq:FG-product} ab=\sum_{g\in G}\biggl(\sum_{h\in G} a_h b_{h^{-1}g}\biggr) g, \quad a,b\in \mathbb{F}_q[G]. \tag*{(2)}\end{align}

Group-algebra code

A group-algebra code is a \( k \)-dimensional linear subspace of the group algebra of \( G\) with coefficients in the field \(GF(q) = \mathbb{F}_q\) with \(q\) elements. To be precise, the code must be closed under permutations corresponding to the elements of the group \( G \); therefore, \( G \) must be a subgroup of the permutation automorphism group of the code, which is defined as the group of permutations of the physical bits that preserve the code space. This leads us to the formal definition of a group-algebra code: a group-algebra code is an ideal in the group algebra \( \mathbb{F}_q G \).

A linear code is a group-algebra code for a group \(G\) if and only if \(G\) is isomorphic to a regular subgroup of the code''s permutation automorphism group [2][3; Thm. 16.4.7].