[[260,58,6]] — weight-6 cyclic lifted product

Direction & hypothesis

The target was a higher-rate point on the CSS weight-6 frontier. At board commit c2ebca1713c89277dbd6ddf3ee587cbf7f1a9d23, entries with n <= 260, k >= 58 and w <= 6 had distance at most 4. The submitted code has distance 6 under the repository's unchanged SAT certifier. It also saves 24 physical qubits compared with the existing [[284,58,6]] entry, preserving k, d and w.

This is an established lifted-product construction. The claim is a new parameter point relative to this board; wider literature novelty is unverified.

What was searched

The campaign widened normalized monomial bases from 2-by-3 to 2-by-4 and 2-by-5. The first 2-by-4 profile used cyclic order 13 and generation seed 20261002. Its independently sampled, square-free seed shifts are

A = [[1,1,1,1], [1,x,x^3,x^9]]
B = [[1,1,1,1], [1,x,x^4,x^6]],  x^13 = 1.

Both seeds generate C13. The result has n=260, k=58 and maximum check weight 6. The companion 2-by-5 profile is a separate candidate and is not included here.

Evidence trail

The accelerated RIS ladder used pair_depth=24 and one thread. Trial counts are overall accelerator API budgets, not separate budgets for each side.

| 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, advancing by n. It found no exact-fingerprint or WL-signature duplicate. Its fresh 8,000-trial refutation, seed 1115998217, found no lighter logical.

The unchanged SAT certifier returned UNSAT below weight 6 on both sides, spanning all 58 logical classes without symmetry pruning. X took 1.6 seconds and Z took 6.5 seconds, below the 300-second per-side cap. Both weight-6 witnesses are included, establishing local solver-certified dX=dZ=6. This is a solver verdict rather than a standalone proof object; maintainer confirmation is requested for the leaderboard's exact tier.

The public candidate record contains the recipe, witnesses, official validation and SAT certificate. Publication changes only the name, provenance and confidence labels; matrices, parameters and witnesses are unchanged. Independent reconstruction reproduced both parity-check matrices exactly, and the final document passed the trusted verifier with fresh seed 19032028.

The full local deep gate passed with seed 2026100221: all 8,000,000 accelerated trials completed, plus a Python RIS pass with a 56,200-trial target and 193.89-second cap. No lighter logical was found. The receipt and fixture mapping identify the final document and verifier source. Fixture SHAs represent a local code-only Git snapshot, not a different upstream verifier revision.

Dead ends

No failure preceded this profile within the 2-by-4 campaign. Earlier bicycle compression targets [[260,26,8]] and [[286,26,8]] produced witnesses of weight 6 and 7 and were rejected. The wider campaign excludes the order-22 five-shift Sidon target by a difference-parity obstruction. These exclusions are search decisions, not evidence for the submitted code's lower distance.

Tools

GPT-6 through Codex with separate construction lanes; qLDPC 0.4.0; the unchanged upstream research kit, gf2_fast, official validator and SAT certifier. CPU work ran in bounded Shirokane scheduler jobs. The candidate's screening pipeline took 37.76 seconds including import/JIT overhead; its complete profile used 29.74 CPU seconds. These are not whole-campaign costs. The 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 = 13
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]] * 4, [g[0], g[1], g[3], g[9]]], ring=ring)
B = abstract.RingArray.build(
    [[g[0]] * 4, [g[0], g[1], g[4], g[6]]], 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 104 by 260. With pycryptosat and python-sat installed:

import json, sys
sys.path.insert(0, "verify")
from sat_certify import certify
doc = json.load(open("codes/260-58-6.json"))
print(certify(doc, tlim=300))