Classical-product code[1][2]

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}