# Group-algebra code

## Description

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

## Notes

## Parent

- 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].

## Children

- 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 \( \mathbb{Z}_n \).

## Cousins

- Binary quadratic-residue (QR) code — The self-dual \([48,24,12]\) extended quadratic residue code is a group-algebra code [7][1; Ex. 16.5.1].
- Golay code — The extended Golay code is a group-algebra code for various groups [8][9][10][1; Ex. 16.5.1].
- Hermitian code — Some Hermitian codes are group-algebra codes [11][1; Remark 16.4.14].
- Klein-quartic code — Some Klein-quartic codes are group-algebra codes [1; Remark 16.4.14].
- Suzuki-curve code — Some Suzuki-curve codes are group-algebra codes [1; Remark 16.4.14].
- Generalized RM (GRM) code — GRM codes over prime-power fields are group-algebra codes [5][6][12][1; Ex. 16.4.11].

## References

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- J. J. Bernal, Á. del Río, and J. J. Simón, “An intrinsical description of group codes”, Designs, Codes and Cryptography 51, 289 (2009) DOI
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- S. D. Berman, “On the theory of group codes”, Cybernetics 3, 25 (1969) DOI
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- Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.
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- I. McLoughlin, “A group ring construction of the [48,24,12] type II linear block code”, Designs, Codes and Cryptography 63, 29 (2011) DOI
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- I. McLoughlin and T. Hurley, “A Group Ring Construction of the Extended Binary Golay Code”, IEEE Transactions on Information Theory 54, 4381 (2008) 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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- Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
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## 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