## Description

2BGA codes are the smallest LP codes LP\((a,b)\), constructed from a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group, and \(\mathbb{F}_q\) is a Galois field. For a group of order \(\ell\), we get a 2BGA code of length \(n=2\ell\). A 2BGA code for an Abelian group is called an Abelian 2BGA code.

An \(\mathbb{F}_q\)-linear code isomorphic to a \(Z\) logical subspace of the 2BGA code LP\((a,b)\) can be most naturally defined as a linear space of pairs \((u,v)\in \mathbb{F}_q[G]\times \mathbb{F}_q[G]\) such that \begin{align} a u+v b=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 \([[24,5,3]]_2\) [2].

## Protection

## Rate

The 2BGA construction gives some of the best short codes with small stabilizer weights.

A number of 2BGA codes \([[n,k,d]]_q\) with row weights \(W\le 8\), block lengths \(n\le 100\), and parameters such that \(kd\ge n\) have been constructed by exhaustive enumeration [2]. Examples include GB codes with parameters \([[70,8,10]]_2\), \([[72,10,9]]_2\), Abelian 2BGA for groups \(\mathbb{Z}_{mh}=\mathbb{Z}_m\times \mathbb{Z}_2\) (index-4 QC codes) with parameters \([[48,8,6]]_2\) and \([[56,8,7]]_2\), and non-Abelian codes with parameters \([[64,8,8]]_2\), \([[82,10,9]]_2\), \([[96,10,12]]_2\), and \([[96,12,10]]_2\) (all of these have stabilizer generators of weight \(W=8\).)

## Parents

- Lifted-product (LP) code — 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.
- Two-block quantum code — 2BGA codes are two-block quantum codes whose commuting matrices are constructed with the help of a group algebra.

## Children

- Bivariate bicycle (BB) code — Bivariate bicycle codes are Abelian 2BGA codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).
- Generalized bicycle (GB) code — A code GB\((a,b)\) with circulants of size \(\ell\) is a special case of a 2BGA code over the cyclic group \(\mathbb{Z}_{\ell}\). More precisely, for the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle \), any element \(a\) of the group algebra \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) can be seen as a polynomial \(a(x)\in \mathbb{F}_q[x]\) over the group generator \(x\), where the polynomial degree \(\deg a(x)<\ell\). The 2BGA code LP\((a,b)\) is then just a generalized bicycle code GB\([a(x),b(x)]\) constructed from the polynomials \(a(x)\) and \(b(x)\) corresponding to \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\).

## Cousins

- Quantum low-density parity-check (QLDPC) code — Given group algebra elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) (i.e., number of non-zero terms in the expansion), the 2BGA code LP\((a,b)\) has stabilizer generators of uniform weight \(W_a+W_b\).
- Quasi-cyclic quantum code — Any Abelian 2BGA code can be thought of as a multi-dimensional index-two quasi-cyclic code. More precisely, any finite Abelian group can be written as a direct product of several cyclic groups, e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product of \(D\) cyclic groups, which is equivalent to a representation \begin{align} G=\langle x_1,\ldots,x_D|x_j^{m_j}=1, x_jx_ix_j^{-1}x_i^{-1}=1, \forall 1\le i,j\le D\rangle. \tag*{(3)}\end{align} Respectively, an element of the group algebra \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is a finite field, can be written as a \(D\)-variate polynomial in \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of order \(m_j\) not exceeding \(m_j-1\). An equivalent construction in terms of Kronecker products of circulant matrices was introduced in [4]. Related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in [5].
- Group-algebra code

## References

- [1]
- R. Wang, H.-K. Lin, and L. P. Pryadko, “Abelian and non-abelian quantum two-block codes”, (2023) arXiv:2305.06890
- [2]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
- [3]
- 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.
- [4]
- A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
- [5]
- S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137

## 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