Group-algebra code


Also known as a group code. 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.

The 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 \( \mathbb{F}_q G \).


See Ch. 16 of Ref. [1] and pg. 58 of Ref. [2] for introductions to group-algebra codes.Not all abelian group-algebra codes are for cyclic groups (cyclic codes) or for elementary abelian \( p \) groups (e.g. Reed Muller codes [3]). For example, there is a binary code with parameters \( [45,13,16] \) which is an abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \).


  • Linear \(q\)-ary code — 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 [4][1; Thm. 16.4.7].


  • Reed-Muller (RM) code — RM codes are group-algebra codes [5][6][1; Ex. 16.4.11]. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the group algebra \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).
  • Cyclic linear \(q\)-ary code — A length-\(n\) cyclic \(q\)-ary linear code is an abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).



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Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
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“Group-algebra code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_group, title={Group-algebra code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Group-algebra code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.