n
172
The submitted JSON records witness-backed upper bounds on both distances. Randomized search is not a lower-bound certificate.
Target: CSS weight-9plus x unrestricted. Explore the rate, blocklength and check-weight tradeoffs of small-factor CSS tensor products. This applies an established construction; literature novelty remains unverified.
Seven tensor instances were built after screening small board seeds and parity factors; all seven passed the original trusted gate and six were retained as non-dominated within the batch. Initial pilots used 500 NumPy trials per side. The confirmation ladder below starts with the subsequent 100,000-trial confirmation and includes the fresh million-trial rung. Final validation uses upstream da1f2c0a1611b215376b249c139f4951e2394418.
Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:
| budget (trials/side) | lightest logical | | ---: | ---: | | 10,000 (NumPy; X/Z seeds 20761001/20761002) | 8 | | 10,000 (NumPy; X/Z seeds 20761003/20761004) | 8 | | 10,000 (NumPy; X/Z seeds 20761005/20761006) | 8 | | 10,000 (NumPy; X/Z seeds 20761007/20761008) | 8 | | 10,000 (NumPy; X/Z seeds 20761009/20761010) | 8 | | 10,000 (NumPy; X/Z seeds 20761011/20761012) | 8 | | 10,000 (NumPy; X/Z seeds 20761013/20761014) | 8 | | 10,000 (NumPy; X/Z seeds 20761015/20761016) | 8 | | 10,000 (NumPy; X/Z seeds 20761017/20761018) | 8 | | 10,000 (NumPy; X/Z seeds 20761019/20761020) | 8 | | 1,000,000 (C++; root seed 21061003; threads=4, pair_depth=16) | 8 |
11 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 8 (upper_bound), Z: d <= 8 (upper_bound).
Ten NumPy batches of 10,000 trials per side used X seeds 20761001, 20761003, through 20761019 and the adjacent even Z seeds; every batch found weight 8 on both sides. The final rung uses 1,000,000 trials per side and 4 threads. Each accelerated call searches both Pauli sides and returns the overall minimum, rather than separate minima. Root seeds differ between calls; within a call the backend uses matching thread-seed offsets across sides. NumPy and C++ trial budgets are not interchangeable. Returned witnesses were saved and checked. No smaller logical was found in these rungs. Both submitted sides remain witness-backed upper bounds.
For the even all-ones parity factor, complementary representatives v and v+1 have disjoint supports. The existing overlap bound (Lemma 4 of arXiv:2609.37231) and tensor upper bound give D_P=2d_P(B). The unchanged repository MILP certifier previously proved both distances of the 20-8-4 seed, with a 3-second limit per logical constraint. This supports D_X=D_Z=8 analytically, separately from the randomized evidence; the submitted JSON remains upper_bound and does not claim server exact certification of this product.
product dominates it in blocklength, encoded dimension and check weight.
noncommutation; reconstructing the general tensor blocks gives commuting checks. Rank conditions affect the homology dimension, not the square-zero differential identity.
Model: GPT-6 (Codex desktop; @nihoniumqc). Repo tooling: research/kit/submit.py, research/kit/surrogate.py, verify/gf2_fast.cpp, verify/certify.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: Python 3.12 on macOS arm64. Fresh million-trial rung: 16.94 seconds. Unchanged challenge C++ accelerator with pybind11 3.0.1.
Input entries are codes/8-6-2.json and codes/20-8-4.json at upstream commit da1f2c0a1611b215376b249c139f4951e2394418. Input attribution: 8-6-2: @MathysRennela; 20-8-4: @EthanFeld. The 6-qubit factor is the known all-ones error-detecting CSS code. The 8- and 12-qubit factors, when used, are board reproductions of the 2BGA construction (arXiv:2306.16400). The 32-qubit seed is a SAT search result, the 20-qubit seed is a self-orthogonal CSS search result, and the 15-qubit seed is the standard quantum Hamming code. Only the two named inputs are used in this submission; their stored layout coordinates are not inherited.
The following code reconstructs the submitted checks, including their ordering:
import json
import numpy as np
def read(name):
with open("codes/" + name) as f:
doc = json.load(f)
result = []
for side in ("X", "Z"):
h = np.zeros((len(doc["checks"][side]), doc["n"]), dtype=np.int8)
for i, support in enumerate(doc["checks"][side]):
h[i, support] = 1
result.append(h)
return result
matrices = read('8-6-2.json') + read('20-8-4.json')
rows = [[0], [0], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]]
hx, hz, gx, gz = [h[r] for h, r in zip(matrices, rows)]
mx, nq = hx.shape
mz = hz.shape[0]
ax, na = gx.shape
az = gz.shape[0]
I = lambda d: np.eye(d, dtype=np.int8)
Z = lambda r, c: np.zeros((r, c), dtype=np.int8)
HX = np.block([
[np.kron(hx, I(na)), Z(mx*na, mz*ax), np.kron(I(mx), gz.T)],
[np.kron(I(nq), gx), np.kron(hz.T, I(ax)), Z(nq*ax, mx*az)]])
HZ = np.block([
[np.kron(hz, I(na)), np.kron(I(mz), gx.T), Z(mz*na, mx*az)],
[np.kron(I(nq), gz), Z(nq*az, mz*ax), np.kron(hx.T, I(az))]])
The X/Z matrices have shapes (68, 172) and (68, 172) and GF(2) ranks 62 and 62. Therefore k=172-62-62=48; maximum check weight is 13. The reconstruction was checked entry-for-entry against the submitted JSON. Use verify/qldpc_verify.py on codes/172-48-8.json to verify the stored logical witnesses and structure. The accelerated search is the witness-returning research/kit/surrogate.py interface distance_rand_witness, backend="fast"; use the seeds, budgets, threads and pair_depth in the ladder. Save the returned support and verify it. There is no layout or circuit-distance claim.