## Description

An LP code for Abelian group \(G\). The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [2; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance.

The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [3]. See Refs. [1,2,4,5] for other explicit examples.

## Rate

Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) [2]; this construction can be derandomized by being reformulated as a balanced product code [6]. Other explicit versions of codes with such parameters have been developed [7].

## Decoding

Ensemble BP decoder for codes without short cycles of length 4 [8].

## Parents

## Children

- Bivariate bicycle (BB) code — Bivariate bicycle codes are Abelian LP 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 lifted-product code LP\((A,B)\) code over the Abelian group algebra \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) associated with a cyclic group, with \(1\times 1\) matrices \(A=a(x)\), \(B=b(x)\) given by the corresponding polynomials. Quasi-cyclic LP codes, i.e., LP codes constructed from cyclic groups, are equivalent to GB codes [2; Sec. III.E].

## Cousins

- Quasi-cyclic LDPC (QC-LDPC) code — QC-LDPC codes can be lifted to yield various Abelian LP codes [4,8,9]. Conversely, the Abelian LP construction yiels notable families of QC-LDPC codes [7].
- Finite-geometry LDPC (FG-LDPC) code — FG-LDPC codes can be used to construct Abelian LP codes [8].
- Expander LP code — Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) [2]; this construction can be derandomized by being reformulated as a balanced product code [6]. Other explicit versions of codes with such parameters have been developed [7].
- Asymmetric quantum code — The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [3].
- Concatenated GKP code — GKP codes have been concatenated with Abelian LP codes [4] that are in turn based on QC-LDPC codes [10].

## References

- [1]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [2]
- P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022) arXiv:2012.04068 DOI
- [3]
- J. Roffe et al., “Bias-tailored quantum LDPC codes”, Quantum 7, 1005 (2023) arXiv:2202.01702 DOI
- [4]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [5]
- Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [6]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
- [7]
- F. G. Jeronimo et al., “Explicit Abelian Lifts and Quantum LDPC Codes”, (2021) arXiv:2112.01647
- [8]
- S. Miao et al., “A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes”, (2024) arXiv:2401.06874
- [9]
- T. R. Scruby, T. Hillmann, and J. Roffe, “High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory”, (2024) arXiv:2406.14445
- [10]
- M. P. C. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices”, IEEE Transactions on Information Theory 50, 1788 (2004) DOI

## Page edit log

- Victor V. Albert (2024-05-06) — most recent

## Cite as:

“Abelian LP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/abelian_lifted_product