Group-algebra code

Group-algebra code[1]

Alternative names: \(G\) code.

Description

An \( [n,k]_q \) code associated with a finite group \(G\) of order \(n\), viewed as an ideal in the group algebra \(\mathbb{F}_q[G]\) [2; Def. 16.4.3]. Equivalently, after identifying the \(n\) coordinate positions of each codeword with elements of \(G\), the code is invariant under the regular action of \(G\) and thus becomes a \(G\)-submodule of \(\mathbb{F}_q^n\) [4][3; 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} Semisimple group algebras can be decomposed into simple components via a Wedderburn-Artin decomposition [6][5; Thm. 4.4, p. 112].

Group-algebra code

A group-algebra code is a \( k \)-dimensional linear subspace of the group algebra of \( G\) with coefficients in \(\mathbb{F}_q\). The formal definition is that a group-algebra code is a left, right, or two-sided 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 [7][2; Thm. 16.4.7].

Notes

See [2; Def. 16.3.1][2; Def. 16.4.3][8; pg. 58] for introductions to group algebras and 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 [9]). 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

\([24, 12, 8]\) Extended Golay code— The extended Golay code is a group-algebra code for various groups [10–12]; see [13][2; Exam. 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 [14][2; Exam. 16.5.1].
Hermitian code— Some Hermitian codes are group-algebra codes [15][2; Remark 16.4.14].
Klein-quartic code— Some Klein-quartic codes are group-algebra codes [2; Remark 16.4.14].
Suzuki-curve code— Some Suzuki-curve codes are group-algebra codes [2; Remark 16.4.14].
Generalized RM (GRM) code— GRM codes over prime-power fields are group-algebra codes [1,16,17][2; Exam. 16.4.11].
Linear code with complementary dual (LCD)— A group code \(C \leq \mathbb{F}_q G\) is LCD if and only if \(C=e \mathbb{F}_q G\) for an idempotent \(e\) satisfying \(e=\hat{e}\), and then \(C^{\perp}=(1-e)\mathbb{F}_q G\) [2; Thm. 16.7.6].
Self-dual linear code— Self-dual group codes exist exactly when the base field has characteristic \(2\) and the underlying group has even order [2; Thm. 16.5.4].
\(q\)-ary simplex code— Over a prime field \(\mathbb{F}_p\), simplex codes with parameters \([(p^m-1)/(p-1),m,p^{m-1}]_p\) and \(\gcd(m,p-1)=1\) are group-algebra codes [2; Exam. 16.8.2].
Divisible code— If \(C\) is a group code over a field of characteristic \(p\), then the monomial kernel \(K_M(C)\) has order dividing the weight of every codeword, and the \(p^{\prime}\)-part of the divisor of \(C\) equals the \(p^{\prime}\)-part of \(|K_M(C)|\) [2; Thm. 16.8.3].
Two-block group-algebra (2BGA) codes— A 2BGA code \(LP(a,b)\) is constructible as a hypergraph-product code when the support subgroups generated by \(a\) and \(b\) are disjoint. In that case, the commuting matrices simultaneously acquire hypergraph-product Kronecker-product form, and the code can be obtained from a pair of classical group-algebra codes [18; Statements 8 and 12].

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

\([8,4,4]\) extended Hamming code

The \([8,4,4]\) extended Hamming code is a group-algebra code for the group \(\mathbb{Z}_2 \times \mathbb{Z}_4\) [13].

Reed-Muller (RM) code

RM codes are group-algebra codes [1,16][2; 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 \( \mathbb{F}_2 G_m \). 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 \(\alpha_j\) [19].

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 [19].

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 \( \mathbb{Z}_n \) [2; Exam. 16.4.9].

References

[1]: S. D. Berman, “On the theory of group codes”, Cybernetics 3, 25 (1969) DOI
[2]: W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[3]: 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
[4]: A. Günther. Automorphism groups of self-dual codes. PhD thesis, Aachen, Techn. Hochsch., 2009, 2009
[5]: K. Doerk and T. O. Hawkes, “Finite Soluble Groups”, (1992) DOI
[6]: M. Sales-Cabrera, X. Soler-Escrivà, and V. Sotomayor, “Codes in algebras of direct products of groups”, (2025) arXiv:2412.09695
[7]: J. J. Bernal, Á. del Río, and J. J. Simón, “An intrinsical description of group codes”, Designs, Codes and Cryptography 51, 289 (2009) DOI
[8]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[9]: S. D. Berman, “Semisimple cyclic and Abelian codes. II”, Cybernetics 3, 17 (1970) DOI
[10]: I. McLoughlin and T. Hurley, “A Group Ring Construction of the Extended Binary Golay Code”, IEEE Transactions on Information Theory 54, 4381 (2008) DOI
[11]: S. T. Dougherty, J. Gildea, R. Taylor, and A. Tylyshchak, “Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings”, (2016) arXiv:1604.07863
[12]: F. Bernhardt, P. Landrock, and O. Manz, “The extended golay codes considered as ideals”, Journal of Combinatorial Theory, Series A 55, 235 (1990) DOI
[13]: M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
[14]: I. McLoughlin, “A group ring construction of the [48,24,12] type II linear block code”, Designs, Codes and Cryptography 63, 29 (2011) DOI
[15]: J. P. Hansen. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987
[16]: P. Charpin. Codes idéaux de certaines algèbres modulaires. PhD thesis, 1982
[17]: P. Landrock and O. Manz, “Classical codes as ideals in group algebras”, Designs, Codes and Cryptography 2, 273 (1992) DOI
[18]: H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
[19]: S. Bhandari, P. Harsha, M. Kumar, and M. Sudan, “Ideal-Theoretic Explanation of Capacity-Achieving Decoding”, LIPIcs, Volume 207, APPROX/RANDOM 2021 207, 56:1 (2021) arXiv:2103.07930 DOI

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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

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

Group-algebra code[1]

Description

Group algebra

Group-algebra code

Notes

Cousins

Member of code lists

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References

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