[[210,56,8]] weight-10 CSS parity amplification

The submitted JSON records witness-backed upper bounds on both distances. Randomized search is not a lower-bound certificate.

Direction & hypothesis

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.

What was searched

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.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 100,000 (C++; root seed 20961001; threads=2, pair_depth=16) | 8 | | 1,000,000 (C++; root seed 21061001; threads=4, pair_depth=16) | 8 |

2 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).

The first accelerated confirmation used 100,000 trials per side, 2 threads and pair_depth=16. 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 32-14-4 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.

Dead ends

• The tested 466-qubit tensor product was excluded because the 414-qubit

product dominates it in blocklength, encoded dimension and check weight.

• A previous amplification fieldnote rejected three products for

noncommutation; reconstructing the general tensor blocks gives commuting checks. Rank conditions affect the homology dimension, not the square-zero differential identity.

Tools

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: 24.46 seconds. Unchanged challenge C++ accelerator with pybind11 3.0.1.

Reproduction

Input entries are codes/6-4-2.json and codes/32-14-4.json at upstream commit da1f2c0a1611b215376b249c139f4951e2394418. Input attribution: 6-4-2: @MathysRennela; 32-14-4: @vprusso. 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('6-4-2.json') + read('32-14-4.json')
rows = [[0], [0], [0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8]]
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 (86, 210) and (86, 210) and GF(2) ranks 77 and 77. Therefore k=210-77-77=56; maximum check weight is 10. The reconstruction was checked entry-for-entry against the submitted JSON. Use verify/qldpc_verify.py on codes/210-56-8.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.