Group-algebra code[1]
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
- Group-orbit code— A group-algebra code admits a regular, i.e., free and transitive, action on coordinates by a subgroup of its permutation automorphism group [2; Thm. 16.4.7]. This differs from a group-orbit code, whose defining group action is transitive on codewords.
- \([24, 12, 8]\) 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].
- \([8,4,4]\) 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 [1,15][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 \( p\neq 2 \).
- Hermitian code— Some Hermitian codes are group-algebra codes [16][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 [1,15,17][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
References
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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