## Description

CSS stabilizer 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 in Ref. [2], can yield QLDPC codes; see Thm. 1 in Ref. [1].

## Parents

- Calderbank-Shor-Steane (CSS) stabilizer code
- Homological product code — Distance balancing relies on taking homological product of chain complexes corresponding to a classical and a quantum code.

## Cousins

- Subsystem qubit stabilizer code
- Balanced product code — Distance balancing is used to form balanced-product subsystem codes [3].
- Fiber-bundle code — Fiber-bundle code constructions use distance balancing to increase distance.
- Ramanujan-complex product code — Ramanujan tensor-product constructions use distance balancing to increase distance.

## Zoo code information

## References

- [1]
- M. B. Hastings, “Weight Reduction for Quantum Codes”. 1611.03790
- [2]
- Shai Evra, Tali Kaufman, and Gilles Zémor, “Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders”. 2004.07935
- [3]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021). DOI; 2012.09271

## Cite as:

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