## 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. These codes have been generalized to the intersecting subset code family [3].

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”, Quantum 8, 1420 (2024) arXiv:2209.13474 DOI
- [3]
- D. Ostrev, “Quantum LDPC Codes From Intersecting Subsets”, IEEE Transactions on Information Theory 70, 5692 (2024) arXiv:2306.06056 DOI

## 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