## Description

A CSS code constructed by separately constructing the \(X\) and \(Z\) check matrices using product constructions from classical codes. A particular \([[512,174,8]]\) code performed well [2] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes.

For example, for \(H_i^x\), \(H_i^z\) being the \(X\) and \(Z\) check matrices of CSS codes \(C_i\) with \(i\in\{1,2,3,4\}\), the 2-fold symmetric classical product code is given by \begin{align} H_{\otimes}^x &:=\left(\begin{array}{c} H_1^x \otimes H_2^x \otimes I \otimes I \\ I \otimes I \otimes H_3^x \otimes H_4^x \end{array}\right) \tag*{(1)}\\ H_{\otimes}^z &:=\left(\begin{array}{c} H_1^z \otimes I \otimes H_3^z \otimes I \\ I \otimes H_2^z \otimes I \otimes H_4^z \end{array}\right)~. \tag*{(2)}\end{align}

## Parents

## Cousins

- Quantum Tanner code — A \([[512,174,8]]\) classical-product code performed well [2] against erasure and depolarizing noise when compared to a member of an asymptotically good quantum Tanner code family.
- Single parity-check (SPC) code — SPC codes are used as component codes in classical-product code constructions.
- Tensor-product code — Tensor-product codes are utilized in classical-product code constructions.

## References

- [1]
- M. Hivadi, “On quantum SPC product codes”, Quantum Information Processing 17, (2018) DOI
- [2]
- D. Ostrev et al., “Classical product code constructions for quantum Calderbank-Shor-Steane codes”, (2022) arXiv:2209.13474

## Page edit log

- Victor V. Albert (2022-11-13) — most recent
- Hengyun (Harry) Zhou (2022-11-13)

## Cite as:

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