# Group-algebra code[1]

## Description

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].

## Notes

## Parents

- Quasi group-algebra code — A quasi group-algebra code of index \(\ell=1\) is a group-algebra code.
- Group-orbit code — A \(q\)-ary group-orbit code hosts a transitive group action. If the action is also free, then the code is a group-algebra code.

## Children

- Reed-Muller (RM) code — RM codes are group-algebra codes [1,6][3; 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 \( p\neq 2 \).
- Multiplicity code — Multiplicity codes of order \(s\) are Abelian group-algebra codes whose corresponding polynomial that is modded out is \((x-\alpha_j)^s\) for each evaluation point \(\alpha_j\) [7].
- Folded RS (FRS) code — FRS codes are polynomial ideal codes whose corresponding polynomial is a product of the polynomials of the RS codes that are being folded [7].
- 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 \( \mathbb{Z}_n \).

## Cousins

- Binary quadratic-residue (QR) code — The self-dual \([48,24,12]\) extended quadratic residue code is a group-algebra code [8][3; Ex. 16.5.1].
- Golay code — The extended Golay code is a group-algebra code for various groups [9–11]; see [3; Ex. 16.5.1].
- \([48,24,12]\) self-dual code — The \([48,24,12]\) self-dual code is a group code for \(G\) being a dihedral group [8][12; Exam. 16.5.1].
- Hermitian code — Some Hermitian codes are group-algebra codes [13][3; Remark 16.4.14].
- Klein-quartic code — Some Klein-quartic codes are group-algebra codes [3; Remark 16.4.14].
- Suzuki-curve code — Some Suzuki-curve codes are group-algebra codes [3; Remark 16.4.14].
- Generalized RM (GRM) code — GRM codes over prime-power fields are group-algebra codes [1,6,14][3; Ex. 16.4.11].
- Two-block group-algebra (2BGA) codes

## References

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- S. Bhandari et al., “Ideal-Theoretic Explanation of Capacity-Achieving Decoding”, (2021) arXiv:2103.07930 DOI
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- S. T. Dougherty et al., “Group rings, G-codes and constructions of self-dual and formally self-dual codes”, Designs, Codes and Cryptography 86, 2115 (2017) DOI
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- F. Bernhardt, P. Landrock, and O. Manz, “The extended golay codes considered as ideals”, Journal of Combinatorial Theory, Series A 55, 235 (1990) DOI
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- E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 177–294.
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- Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
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- P. Landrock and O. Manz, “Classical codes as ideals in group algebras”, Designs, Codes and Cryptography 2, 273 (1992) DOI

## Page edit log

- Victor V. Albert (2022-01-03) — most recent
- Victor V. Albert (2022-11-18)
- Ian Teixeira (2021-12-19)

## Cite as:

“Group-algebra code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group