# Group code

## Description

An \( [n,k] \) binary or \(q\)-ary code based on a finite group \( G \) of size \(n \). A group code for an abelian group is called an abelian group 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 code: a group code is an ideal in the group algebra \( \mathbb{F}_q G \).

## Notes

## Parent

## Children

- Cyclic linear \(q\)-ary code — A length-\(n\) cyclic \(q\)-ary linear code is an abelian group code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).
- Cyclic linear binary code — A length-\(n\) cyclic binary linear code is an abelian group code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).
- Reed-Muller (RM) code — 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 \( p\neq 2 \).

## References

## Page edit log

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

## Zoo code information

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/group.yml.