## Description

Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that is treats both types of errors on a more equal footing. The original distance-balancing procedure [1], later generalized [2; Thm. 4.2], can yield QLDPC codes [1; Thm. 1].

A related procedure called weight reduction [1] takes in a CSS stabilizer code and outputs another CSS code that admits a set of stabilizer generators whose weight is independent of the number of qubits \(n\).

## Parents

## Cousins

- Homological product code — Distance balancing relies on taking a homological product of chain complexes corresponding to a classical and a quantum code.
- Subsystem qubit stabilizer code
- Quantum locally testable code (QLTC) — Distance balancing is useful for constructing QLTCs. Scaling of the soundness of a given code family is proven in [1; Lemma 7] for the original distance balancing scheme and in [3; Thm. 1.1] for the generalized scheme [2].
- Quantum check-product code — Quantum check-product code constructions use distance balancing to increase distance.
- Fiber-bundle code — Fiber-bundle code constructions use distance balancing to increase distance.
- High-dimensional expander (HDX) code — Ramanujan tensor-product constructions use distance balancing to increase distance.
- Balanced product (BP) code — Distance balancing is used to form balanced-product subsystem codes [4].

## References

- [1]
- M. B. Hastings, “Weight Reduction for Quantum Codes”, (2016) arXiv:1611.03790
- [2]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [3]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “General Distance Balancing for Quantum Locally Testable Codes”, (2023) arXiv:2305.00689
- [4]
- 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 (2022-01-20) — most recent

## Cite as:

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