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 verify kernel membership and exclusion from the corresponding 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 \(p_{\mathrm{DE}}\approx0.0733\).