This page distinguishes the two 64-fold lifts discussed 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 selected instance is P64_avoid_w16_orbit.
Its coefficients (fer_distance/data/fer.csv), full matrices, and verification records
are supplied among the version-3 ancillary files.
Section 3.5 explains the correction.
The local coefficient files below are the earlier instance. The CSV SHA-256 is
137f98b600969582691a6c1912ee982e0f541c39ecdc3950cac6a0448493ebae,
matching the file identified in Section 3.5. The original summary and README are preserved.
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 \[ \begin{aligned}a^{(0)}&=(0,1,2),\\b^{(0)}&=(7,3,6),\end{aligned} \] \[ \begin{aligned}a^{(1)}&=(8,13,2),\\b^{(1)}&=(11,10,6).\end{aligned} \] 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 \[ [[10240,4108,16]] \] 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 | This exclusion holds for the selected FER lift. The archived coefficient instance instead contains a weight-16 logical representative in that orbit. |
| Distance lower bound | For this archived lift, version 3 establishes \(d_X\ge18\) and \(d_Z\ge16\). |
| Distance upper bound | A weight-16 nontrivial \(Z\)-logical representative establishes \(d=d_Z=16\) for the archived lift. |
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 earlier paper version listed the following supports for this archived instance:
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.
These measurements use the selected FER-experiment matrices, not the archived coefficient files above. The reported FER data use joint log-domain sum-product BP 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 \(p_{\mathrm{DE}}\approx0.0733\).