# Fiber-bundle code[1]

## Description

Also called a twisted product code. CSS code constructed by combining a random LDPC code as the base and a cyclic repetition code as the fiber of a fiber bundle. After applying distance balancing, a QLDPC code with distance \(\Omega(n^{3/5}\text{polylog}(n))\) and rate \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.

## Rate

Rate \(k/n = \Omega(n^{-2/5}/\text{polylog}(n))\), distance \(d=\Omega(n^{3/5}/\text{polylog}(n))\). This is the first QLDPC code to achieve a distance scaling better than \(\sqrt{n}~\text{polylog}(n)\).

## Decoding

Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [1].

## Parent

- Balanced product code — Fiber-bundle codes can be formulated in terms of a balanced product [2].

## Child

- Homological product code — Fiber-bundle code can be viewed as a homological product code with a twisted product.

## Cousins

- Distance-balanced code — Fiber-bundle code constructions use distance balancing to increase distance.
- Random quantum code

## References

- [1]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber Bundle Codes: Breaking the \(N^{1/2} \operatorname{polylog}(N)\) Barrier for Quantum LDPC Codes”, (2020) arXiv:2009.03921
- [2]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI

## Page edit log

- Victor V. Albert (2022-01-04) — most recent
- Jon Nelson (2021-12-15)

## Cite as:

“Fiber-bundle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fiber_bundle