\(q\)-ary group code

Description

An \( [n,k]_{q} \) 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

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

Parent

Cousins

  • Cyclic code — A length-\(n\) cyclic 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 \( p\neq 2 \).

Zoo code information

Internal code ID: group

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Zoo Code ID: group

Cite as:
“\(q\)-ary group code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group
BibTeX:
@incollection{eczoo_group, title={\(q\)-ary group code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group} }
Permanent link:
https://errorcorrectionzoo.org/c/group

References

[1]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
[2]
S. D. Berman, “Semisimple cyclic and Abelian codes. II”, Cybernetics 3, 17 (1970). DOI

Cite as:

“\(q\)-ary 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/q-ary_digits/group.yml.