This page collects construction information for the \([[10240,4108,10\le d\le32]]\) CSS code in A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases by Koki Okada and Kenta Kasai. The paper is available as arXiv:2605.23894.
The lifted code starts from the \((3,10)\)-regular two-branch base over \(\mathbb F_{16}\). Here \(J=3\), \(|M|=5\), and \(L=2|M|=10\). The base has length \(160\), \(48\) rows on each side, and base CSS parameters \([[160,76,4]]\).
The columns are indexed by \[ (\lambda,t,h)\in\{0,1\}\times\mathbb F_{16}\times M. \] The selected coefficient arrays are \[ a^{(0)}=(0,1,2),\quad b^{(0)}=(7,3,6), \] \[ a^{(1)}=(8,13,2),\quad b^{(1)}=(11,10,6). \] The base has \(720\) \(X\)-\(Z\) row pairs sharing two columns, and \(800\) same-type simple base 6-cycles on each side.
A \(P=64\) CPM lift gives a code of length \[ 160\cdot64=10240 \] with \(3072\) \(X\)-checks and \(3072\) \(Z\)-checks. The computed binary ranks are \[ \operatorname{rank}(H_X^{\mathrm{lift}}) = \operatorname{rank}(H_Z^{\mathrm{lift}}) = 3066, \] so the lifted code has parameters \[ [[10240,4108,\,10\le d\le32]] \] and effective rate \(4108/10240=0.401171875\).
The complete \(P=64\) CPM lift is specified by 960 coefficients, one for each nonzero base edge
of \(H_X\) and \(H_Z\).
In the coefficient files, each record has fields
side, base_row, base_col, and shift_mod_64.
For a base edge \((i,j)\) on the given side with shift \(s\), the lifted row \(64i+r\)
has a nonzero entry in column
\[
64j+((r+s)\bmod 64),\qquad r=0,\ldots,63.
\]
View the full table at p64_lift_coefficients.html, or download it as CSV / JSON.
| Orthogonality after lifting | All 720 paired \(X\)-\(Z\) base overlaps satisfy the zero congruence constraints on CPM exponents. |
|---|---|
| Same-type girth condition | No same-type base 6-cycle has zero signed CPM-exponent sum; the lifted same-type Tanner graphs therefore have girth at least eight. |
| Weight-16 orbit exclusion | The specified two-point lift-coordinate coset pattern with \(K=\{0,32\}\le\mathbb Z/64\mathbb Z\) is excluded for all 20 base supports in the orbit. |
| Distance lower bound | Complete support enumeration found no nonzero vector of weight at most 9 in either lifted kernel, giving \(d_X,d_Z,d\ge10\). |
| Distance upper bound | Explicit verified non-stabilizer logical representatives of weight 32 give \(d_X,d_Z,d\le32\). |
Lifted columns are written as \((c,f)\), meaning column \(64c+f\). For the upper-bound witnesses, let \(K=16\mathbb Z/64\mathbb Z=\{0,16,32,48\}\). The two base-coordinate lists used in the paper are
Expanding \((c,r)\) to \(\{(c,r+k):k\in K\}\) gives weight-32 supports. Direct binary checks on the published coefficient table give zero syndrome on both sides and rank increases (3066\to3067\) after appending the corresponding representative, verifying exclusion from the stabilizer row space.
The reported FER data use joint log-domain belief propagation with deterministic post-processing. At depolarizing probability \(p=0.058\), the run used \(180{,}000{,}000\) trials. There were 25 recorded failures before post-processing and 18 after post-processing, giving \[ 18/180{,}000{,}000=1.0\times10^{-7}. \] The hashing-bound reference for the effective rate is \(p_{\mathrm{hash}}=0.09403285\), and the approximate regular \((3,10)\) BP density-evolution reference is \(p_{\mathrm{DE}}\approx0.0733\).