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Group-algebra code[1]

Alternative names: \(G\) code.

Description

An \( [n,k]_q \) code whose automorphism group includes a finite group \( G \) of size \(n \), which acts on the code via its regular representation. This makes the code a \(G\)-submodule of the module \(GF(q)^n\) [3][2; Lemma 2.3]. 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 [4][5; Thm. 16.4.7].

Notes

See [5][6; pg. 58] 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 [7]). For example, there is a binary code with parameters \( [45,13,16] \) which is an Abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \).

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Quasi group-algebra code
Parents
Quasi group-algebra code
A quasi group-algebra code of index \(\ell=1\) is a group-algebra code.
Group-orbit codeECC
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.
Group-algebra code
Children
\([24, 12, 8]\) Extended Golay code
The extended Golay code is a group-algebra code for various groups [12–14]; see [5; Exam. 16.5.1].
Reed-Muller (RM) code
RM codes are group-algebra codes [1,10][5; Exam. 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 codeGRM
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 \(\alpha_j\) [15].
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 [15].
Cyclic linear \(q\)-ary codeQR
A length-\(n\) cyclic \(q\)-ary linear code is an Abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).

References

[1]
S. D. Berman, “On the theory of group codes”, Cybernetics 3, 25 (1969) DOI
[2]
A. Günther and G. Nebe, “Automorphisms of doubly even self-dual binary codes”, Bulletin of the London Mathematical Society 41, 769 (2009) arXiv:0810.3787 DOI
[3]
Günther, Annika. Automorphism groups of self-dual codes. Diss. Aachen, Techn. Hochsch., Diss., 2009, 2009.
[4]
J. J. Bernal, Á. del Río, and J. J. Simón, “An intrinsical description of group codes”, Designs, Codes and Cryptography 51, 289 (2009) DOI
[5]
W. Willems, "Codes in Group Algebras." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[6]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[7]
S. D. Berman, “Semisimple cyclic and Abelian codes. II”, Cybernetics 3, 17 (1970) DOI
[8]
I. McLoughlin, “A group ring construction of the [48,24,12] type II linear block code”, Designs, Codes and Cryptography 63, 29 (2011) DOI
[9]
Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
[10]
Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.
[11]
P. Landrock and O. Manz, “Classical codes as ideals in group algebras”, Designs, Codes and Cryptography 2, 273 (1992) DOI
[12]
I. McLoughlin and T. Hurley, “A Group Ring Construction of the Extended Binary Golay Code”, IEEE Transactions on Information Theory 54, 4381 (2008) DOI
[13]
S. T. Dougherty, J. Gildea, R. Taylor, and A. Tylyshchak, “Group rings, G-codes and constructions of self-dual and formally self-dual codes”, Designs, Codes and Cryptography 86, 2115 (2017) DOI
[14]
F. Bernhardt, P. Landrock, and O. Manz, “The extended golay codes considered as ideals”, Journal of Combinatorial Theory, Series A 55, 235 (1990) DOI
[15]
S. Bhandari, P. Harsha, M. Kumar, and M. Sudan, “Ideal-Theoretic Explanation of Capacity-Achieving Decoding”, (2021) arXiv:2103.07930 DOI
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Zoo Code ID: group

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/group/group.yml.