This page collects construction information for the \([[16384,4142,18\le d\le32]]\) CSS code in High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures by Koki Okada and Kenta Kasai. The paper is available as arXiv:2604.20838. The decoder implementation and the bundled P=32 code data are also available at kasaikenta/quantum-ldpc-decoder.
The files below preserve the original construction archive. Its summary and README may retain the earlier \(d\le40\) statement; current distance certificates accompany version 4 on arXiv.
The lifted code starts from a length-512 affine-coset CSS base. The base matrices satisfy \[ H_X^{\mathrm{base}},H_Z^{\mathrm{base}}\in\mathbb F_2^{192\times512}. \] Every row has weight \(8\), every column has weight \(3\), and both Tanner graphs have girth \(8\). The associated base CSS code has parameters \[ [[512,174,8]]. \]
The base is permutation-equivalent to the SPC(3) product CSS code. The affine-coset description in the paper gives a direct finite-length proof of CSS orthogonality, \((3,8)\)-regularity, and absence of same-type 4- and 6-cycles.
A \(P=32\) CPM lift gives a code of length \[ 512\cdot32=16384 \] with \(6144\) \(X\)-checks and \(6144\) \(Z\)-checks. Row reduction over \(\mathbb F_2\) gives \[ \begin{aligned} \operatorname{rank}(H_X^{\mathrm{lift}})&=6121,\\ \operatorname{rank}(H_Z^{\mathrm{lift}})&=6121,\\ 192\cdot32-23&=6121. \end{aligned} \] so the lifted code has \[ k=16384-2\cdot6121=4142 \] encoded qubits and parameters \[ [[16384,4142,18\le d\le32]]. \] These are certified lower and upper bounds from version 4; the exact distance is not determined.
The complete \(P=32\) CPM lift is specified by 3072 coefficients, one for each nonzero base edge
of \(H_X\) and \(H_Z\).
In the normalized coefficient files, each record has fields
side, base_row, base_col, and shift_mod_32.
For a base edge \((i,j)\) on the given side with shift \(s\), the lifted row \(32i+r\)
has a nonzero entry in column
\[
32j+((r+s)\bmod 32),\qquad r=0,\ldots,31.
\]
View the full table at p32_lift_coefficients.html, or download it as CSV / JSON. The original repository labels are also preserved as cpm_labels.csv.
| Base regularity and girth | The base is \((3,8)\)-regular and both same-type Tanner graphs have girth exactly \(8\). |
|---|---|
| Orthogonality after lifting | The CPM labels satisfy the local congruence condition for every two-point \(X\)-\(Z\) row intersection, preserving CSS orthogonality. |
| Lifted matrix files | The lifted \(H_X\) and \(H_Z\) matrices are published in MatrixMarket coordinate format as cpm_HX.mtx and cpm_HZ.mtx. |
| Current distance bounds | Version 4 proves \(18\le d_X,d_Z\le32\). The older weight-40 residual is a separate decoder observation, superseded as the best upper bound. |
The short lift report bundled with the code data records \(18816\) base 8-cycles on each side. For the selected CPM labels, the reported zero-voltage base 8-cycles are \(1466\) for \(X\) and \(1491\) for \(Z\), giving \(46912\) lifted \(X\)-side 8-cycles and \(47712\) lifted \(Z\)-side 8-cycles.
The reported FER data use belief propagation with post-processing under the code-capacity depolarizing model. At depolarizing probability \(p=0.085\), the run observed \(23\) final failures in \(2{,}310{,}469{,}200\) trials, giving FER on the \(10^{-8}\) scale. Of these failures, one satisfied the syndrome after post-processing but was not stabilizer-equivalent to the true error; its \(Z\)-residual had weight \(40\), giving the decoder-derived \(d\le40\) evidence. The regular \((3,8)\) density-evolution reference plotted in the paper is \(p_{\mathrm{DE}}\simeq0.1009\).