# Expander lifted-product code[1]

## Description

Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs. For certain parameters, this construction yields the first asymptotically good QLDPC codes [1].

An expander lifted-product code family is constructed as follows. First, take the Cayley graph of a finite group \(G\). Second, take the double cover of the graph, resulting in a graph that satisfies the requirements of participating in a \(G\)-lifted product (i.e., the resulting graph is a free \({\mathbb{F}}_q G\)-module). Third, create a Tanner code out of the graph, in which parity-check supports are defined by the graph, and bitstrings satisfying a particular parity check are defined to be the codewords of a small classical code (chosen to be a random code in the construction). Fourth, take the \(G\)-lifted product of two copies of the Tanner code.

The small classical codes used in the construction of good QLDPC codes are required to have a certain product-expansion property (Lemma 10 in Ref. [1]); it is proven that random codes satisfy said property in the thermodynamic limit.

## Protection

## Rate

## Notes

## Parent

- Lifted-product (LP) code — Lifted products of certain classical Tanner codes are the first asymptotically good QLDPC codes.

## Cousins

- Tanner code — Expander lifted-product codes are products of Tanner codes defined on expander graphs.
- Random code — Expander lifted-product codes are quantum CSS codes that utilize short classical codes in their construction which need to satisfy some properties (Ref. [1], Lemma 10). It is shown that such codes exist, but they are not explicitly constructed. Such codes can be obtained by repeated random sampling or by performing a search of all codes of desired length. Nevertheless, since the length of the desired short codes does not scale with \(n\), this construction is effectively explicit.
- Quantum Tanner code — Quantum Tanner codes are an attempt to construct asymptotically good QLDPC codes that are similar to but simpler than expander lifted-product codes.

## Zoo code information

## References

- [1]
- Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
- [2]
- P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022). DOI; 2012.04068
- [3]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021). DOI; 2012.09271
- [4]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309

## Cite as:

“Expander lifted-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/expander_lifted_product