Description
2BGA codes are the one-by-one, or smallest, LP codes: \(LP(a,b)\) is defined by a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group. If \(|G|=\ell\), then the code has length \(n=2\ell\).
A cyclic group \(G\) yields a GB code, while an Abelian group yields an Abelian 2BGA code [4], i.e., a quasi-Abelian analog of an index-two quasi-cyclic code. In the special case where the support subgroups generated by \(a\) and \(b\) are disjoint, the code reduces to a square-matrix hypergraph-product code built from a pair of classical group-algebra codes [3].
A \(Z\)-type logical subspace of \(LP(a,b)\) can be represented by pairs \((u,v)\in \mathbb{F}_q[G]\times \mathbb{F}_q[G]\) satisfying \begin{align} au+vb=0, \tag*{(1)}\end{align} with any two pairs \((u,v)\) and \((u',v')\) such that \(u'=u+w b\) and \(v'=v-aw\) considered equivalent. The order in the products is relevant when the group is non-Abelian.
For example, consider the alternating group \(G=A_4=T\), also known as the rotation group of a regular tetrahedron, \begin{align} T=\langle x,y|x^3=(yx)^3=y^2=1\rangle,\quad |T|=12, \tag*{(2)}\end{align} and the binary algebra \(\mathbb{F}_2[T]\). Select \(a=1+x+y+x^{-1}yx\) and \(b=1+x+y+yx\) to get an essentially non-Abelian 2BGA code LP\([a,b]\) with parameters \([[24,5,3]]\) [3].
Protection
Some upper and lower bounds on parameters and many examples of 2BGA codes are given in Ref. [3]. The code dimension \(k\) for Abelian 2BGA codes is always even [4].Rate
The 2BGA construction gives some of the best short codes with small stabilizer weights.
Exhaustive enumeration in [3] covered inequivalent connected 2BGA codes with row weight \(W\le 8\), up to length \(n\le 100\) for Abelian groups and \(n\le 200\) for non-Abelian groups. Among the resulting codes with \(kd\ge n\) are GB codes with parameters \([[70,8,10]]\) and \([[72,10,9]]\), Abelian 2BGA codes over groups such as \(\mathbb{Z}_m\times \mathbb{Z}_2\) (index-four quasi-cyclic codes) with parameters \([[48,8,6]]\) and \([[56,8,7]]\), and non-Abelian codes with parameters \([[64,8,8]]\), \([[82,10,9]]\), \([[96,10,12]]\), and \([[96,12,10]]\). All of these examples have stabilizer generators of weight \(W=8\).
Gates
Qubit 2BGA codes can admit a cup product structure and can thus have logical gates in the Clifford hierarchy implemented by constant-depth Clifford circuits [7].Notes
The idea of generating stabilizers according to a Cayley graph of a group was independently proposed by Hastings [1]; see Ref. [8], which contains an error in Prop. 3.2 [9].A database of 2BGA codes is available at QECDB by S. Burton.Cousins
- QLDPC code— Given group algebra elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) (i.e., number of nonzero terms in the expansion), the 2BGA code LP\((a,b)\) has stabilizer generators of uniform weight \(W_a+W_b\).
- Group-algebra code— 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 [3; Statements 8 and 12].
- Hypergraph product (HGP) code— 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 [3; Statements 8 and 12].
- Abelian LP code— Abelian 2BGA codes are LP\((a,b)\) codes, constructed from a pair of one-by-one matrices \(a,b\in \mathbb{F}_q[G]\) in a group algebra of an Abelian group \(G\).
- Affine-permutation-matrix LDPC (APM-LDPC) code— APM-LDPC codes can be used to construct non-Abelian 2BGA codes based on the affine permutation group [10].
Primary Hierarchy
References
- [1]
- M. B. Hastings, LR codes, private communication, 2014.
- [2]
- R. Wang, H.-K. Lin, and L. P. Pryadko, “Abelian and non-abelian quantum two-block codes”, (2023) arXiv:2305.06890
- [3]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
- [4]
- Kalachev, G. V., and Panteleev, P. A. (2020). On the minimum distance in one class of quantum LDPC codes. Intelligent systems. Theory and applications, 24(4), 87-117.
- [5]
- J. N. Eberhardt and V. Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, (2024) arXiv:2407.03973
- [6]
- H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and Dan E Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, (2025) arXiv:2409.18175
- [7]
- N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2025) arXiv:2410.16250
- [8]
- K. T. Tian, E. Samperton, and Z. Wang, “Haah codes on general three-manifolds”, Annals of Physics 412, 168014 (2020) arXiv:1812.02101 DOI
- [9]
- Hsiang-Ku Lin and Leonid Pryadko, private communication, 2024.
- [10]
- K. Kasai, “Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction”, (2025) arXiv:2504.17790
Page edit log
- Victor V. Albert (2023-10-16) — most recent
- Leonid Pryadko (2023-10-10)
Cite as:
“Two-block group-algebra (2BGA) codes”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/2bga