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[[16,6,4]] d =
n
16
k
6
d
4
kd²/n
6.0
w
8
g
0.359
r
2.8586
layers
1

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Distance

d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[1, 6, 8, 15]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[1, 6, 8, 15]
certificate exact, d = 4 · scipy/HiGHS MILP
X: no logical < 4 exists; Z: no logical < 4 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.859
check (X = Z, self-dual)qubit site (16)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction CSS code from Reed-Muller RM(1,4): H_X = H_Z = the RM(1,4) generator with the all-ones row dropped, which halves the check weight from 16 to 8 while generating the same stabilizer group. The submitted artifact is the LAYOUT: 16 qubits placed in the plane by direct minimization of the maximum check diameter subject to unit minimum site spacing, reaching r = 2.8586.
model Claude Claude Opus 5 (claimed, not verified)
builds on quant-ph/9601029 (Steane, 1996), errorcorrectionzoo.org/c/quantum_reed_muller
date 2026-07-25
notes The CODE is textbook and its parameters are known (Steane 1996 / Reed-Muller CSS constructions); provenance.novelty is known_parameters. The contribution is the verified single-layer 2D-local LAYOUT, which is what the board's geometric efficiency g = 4kd^2/(n rho^2 r^4) measures. At r = 2.8586, rho = 1 this scores g = 0.3594 versus 0.0717 for the previous best qLDPC entry on the board. Distance is MILP-certified exact on both sides (verify/certify.py: no logical < 4 exists), filed as upper_bound per board policy. CAVEAT: this radius is NOT converged - independent seeds landed at 2.8586 / 2.8593 / 2.8693 / 2.8773 / 2.8794, so a better layout very likely exists. The per-check packing floor for weight-8 is 2.2470, which would give g = 0.944; the gap is open. RM(1,4) also has basis freedom (its stabilizer in GL(5,2) is only AGL(4,2), giving four basis classes with different check-pair graphs), worth roughly 0.09 of radius and not fully explored.
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[16,6,4]] — CSS Reed–Muller RM(1,4), single-layer layout at r = 2.8586

Direction & hypothesis

Target: the weight-8 × local-2d-single cell (empty before this and the sibling [[15,7,3]] submission). Same hypothesis as the sibling: under g = 4kd²/(nρ²r⁴), a small textbook code with an optimized layout beats large qLDPC entries. The code is Steane's [[16,6,4]]; only the layout is new.

What was searched

  • Check basis first: H_X = H_Z = the RM(1,4) generator **with the all-ones

    • row dropped**, halving check weight 16 → 8 while generating the same stabilizer group. Without this reduction no small-radius layout exists.

  • Layout: minimize max check diameter subject to unit min site spacing over

    • free point positions (lattice-constrained layouts were measurably worse). Multiple independent optimizer seeds; basin hopping from incumbents.

Evidence trail

  • Distance: MILP-certified exact via verify/certify.py ("no logical < 4

    • exists"); filed upper_bound per board policy. Independently re-verified exact (d_X = d_Z = 4) by exhaustive kernel enumeration on 2026-07-27.

  • Radius: not converged. Independent seeds landed at 2.8586 / 2.8593 /

    • 2.8693 / 2.8773 / 2.8794 — a genuine spread, unlike the sibling's clean convergence to √7. Read g = 0.3594 as a floor, not a limit.

Dead ends

  • Same multistart trap as [[15,7,3]]: random restarts return a consistent

    • false optimum; only seeded basin hopping made progress.

  • The weight-8 per-check packing floor (2.2470, worth g ≈ 0.944) was not

    • approached; it may not be simultaneously satisfiable across overlapping checks.

Open leads

RM(1,4) has basis freedom: its stabilizer in GL(5,2) is only AGL(4,2), giving four check-basis classes with different check-pair graphs (90/90/93/93 constrained pairs). Edge count does not predict the winner, so all four need laying out — plausibly worth ~0.09 of radius.

Tools

Claude Opus 5 (matches provenance.model), layout-optimization campaign of 2026-07-25; verify/certify.py, verify/qldpc_verify.py. Minutes per optimizer run at n = 16.

Reproduction

H_X = H_Z = RM(1,4) generator minus the all-ones row; supports in codes/16-6-4.json, layout in locality.coordinates. Verify with uv run python verify/qldpc_verify.py codes/16-6-4.json.

Parity checks

X-checks 5 · Z-checks 5
H_X (5 checks, sparse supports)
[1, 3, 5, 7, 9, 11, 13, 15] [0, 2, 4, 6, 8, 10, 12, 14] [2, 3, 6, 7, 10, 11, 14, 15] [4, 5, 6, 7, 12, 13, 14, 15] [8, 9, 10, 11, 12, 13, 14, 15]
H_Z (5 checks, sparse supports)
[1, 3, 5, 7, 9, 11, 13, 15] [0, 2, 4, 6, 8, 10, 12, 14] [2, 3, 6, 7, 10, 11, 14, 15] [4, 5, 6, 7, 12, 13, 14, 15] [8, 9, 10, 11, 12, 13, 14, 15]
raw submission JSON