[[416,39,6]] — weight-5 cyclic lifted product

Direction & hypothesis

The target is the CSS frontier at low check weight. At board commit c2ebca1713c89277dbd6ddf3ee587cbf7f1a9d23, no entry with n <= 416, k >= 39 and w <= 5 has d >= 4. The submitted code has d = 6 under a local run of the repository's unchanged SAT certifier. It also improves our initial [[455,39,6]] construction by saving 39 physical qubits while preserving capacity, distance and check weight.

This is an established lifted-product construction, not a claim of a new code family or an exhaustive literature record.

What was searched

The improvement campaign allowed either normalized 2-by-3 seed to generate a proper subgroup, while requiring the two subgroup indices to be coprime. For cyclic order N and indices gA,gB, the dimension is k=N+gA+gB+2 gcd(gA,gB). This makes a smaller block with the same dimension possible. Generation seed: 20261002. The selected recipe is

A = [[1, 1, 1], [1, x^4, x^24]]
B = [[1, 1, 1], [1, x^9, x^11]],  x^32 = 1.

Here gA=4 and gB=1. Both full-code connectivity checks pass. The code has 416 physical qubits, 39 logical qubits, and maximum check weight 5.

Evidence trail

The accelerated RIS ladder used pair_depth=24 and one thread. These are overall accelerator API trial counts, not per-side counts.

| Trials | Seed | Retained upper bound | |---:|---:|---:| | 300 | 1116998189 | 6 | | 5,000 | 1117998192 | 6 | | 50,000 | 1118998195 | 6 |

The official validator returned passed=true and board_advancing=true for CSS weight-6 x unrestricted, with no exact-fingerprint or WL-signature duplicate. Its fresh 8,000-trial refutation used seed 1115998217 and found no lighter logical. Relative to the upstream snapshot, this is an undominated tradeoff; no existing entry is strictly dominated and there is no d-only peer.

The unchanged SAT certifier returned UNSAT below weight 6 on both sides, spanning all 39 logical classes without symmetry pruning. X took 4.2 seconds and Z took 6.4 seconds under a 300-second per-side cap. Both weight-6 logical witnesses are included, so the local solver results establish dX=dZ=6. This is a solver certificate, not a standalone proof object. Maintainer confirmation is requested for the leaderboard's exact tier.

The public search record contains the recipe, screening records and witnesses. Its official validator receipt and SAT certificate refer to the provisional candidate document. The submitted matrices, parameters and witnesses are identical; its name, provenance and confidence labels were updated for publication.

The final publication JSON was reconstructed independently: both parity-check matrices matched the recorded recipe exactly, and the trusted verifier passed with fresh seed 19032027. The full local frontier gate also passed using seed 2026100217: all 8,000,000 accelerated trials completed, plus a Python RIS pass with a 74,920-trial target and 240-second cap. Neither found a lighter logical. The deep-gate receipt, snapshot mapping, and matrix reconstruction record these checks. The fixture SHAs identify a local code-only Git fixture; the verifier source is the unchanged upstream commit stated above. Both publication jobs have terminal scheduler accounting with failed=0 and exit=0.

Dead ends

The campaign retained failed targets and their witnesses. For example, paired bicycle compressions to [[260,26]] and [[286,26]] found logicals of weight 6 and 7 respectively, below the needed distance 8. They were rejected. A promised distance or a screen surviving refutation is never counted as an exact discovery.

Tools

GPT-6 through Codex with separate construction lanes; qLDPC 0.4.0; the unchanged upstream research kit, gf2_fast accelerator, validator and SAT certifier. Computation used bounded CPU jobs on Shirokane. The selected candidate's screening pipeline took 53.42 seconds including import/JIT overhead; this is not total campaign compute. A known [[72,12,6]] control and all 11 upstream SAT regression tests passed. No circuit-level or logical-error-rate performance is claimed.

Reproduction

With qLDPC 0.4.0, this constructs the submitted check ordering exactly:

import numpy as np
from qldpc import abstract, codes

m = 32
table = (np.arange(m)[:, None] + np.arange(m)[None, :]) % m
group = abstract.Group.from_table(table)
g = list(group.generate())
ring = abstract.GroupRing(group, field=2)
A = abstract.RingArray.build(
    [[g[0], g[0], g[0]], [g[0], g[4], g[24]]], ring=ring)
B = abstract.RingArray.build(
    [[g[0], g[0], g[0]], [g[0], g[9], g[11]]], ring=ring)
code = codes.LPCode(A, B, set_logicals=False)
HX = np.asarray(code.matrix_x, dtype=np.uint8)
HZ = np.asarray(code.matrix_z, dtype=np.uint8)

Each matrix is 192 by 416. With pycryptosat and python-sat installed, run the upstream exact check:

import json, sys
sys.path.insert(0, "verify")
from sat_certify import certify
doc = json.load(open("codes/416-39-6.json"))
print(certify(doc, tlim=300))