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[[20,8,4]] d =
n
20
k
8
d
4
kd²/n
6.4
w
10
X/Z
1
g
0.151
r
3.6056
layers
1
swaps
20

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[0, 3, 6, 11]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[1, 2, 10, 12]
certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT
X: no logical < 4 exists; Z: no logical < 4 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8–10 (mean 8.333) · H_Z 8–10 (mean 8.333)
qubit degrees H_X 1–5 (mean 2.5) · H_Z 1–5 (mean 2.5)
trapping sets H_X (1,1)×3 (2,1)×22 (3,1)×99 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 3 (1,2): 8 (1,3): 6 (1,4): 2 (1,5): 1 (2,1): 22 (2,2): 33 (2,3): 44 (2,4): 28 (2,5): 6 (3,1): 99 (3,2): 262 (3,3): 264 (3,4): 133 (3,5): 57 (3,6): 25
trapping sets H_Z (1,1)×3 (2,1)×22 (3,1)×99 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 3 (1,2): 8 (1,3): 6 (1,4): 2 (1,5): 1 (2,1): 22 (2,2): 33 (2,3): 44 (2,4): 28 (2,5): 6 (3,1): 99 (3,2): 262 (3,3): 264 (3,4): 133 (3,5): 57 (3,6): 25
witness diameter X 4.4721 · Z 3.1623 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 3.606
check (X = Z, self-dual)qubit site (20)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
routing cost 20 nearest-neighbor SWAPs per round in total, at most 2 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @EthanFeld
provenance submitted through the challenge
novelty novelty not audited
construction Randomized self-orthogonal CSS search: n=20, stabilizer rank 6, H_X = H_Z. Column labels form a 6-bit cap (all first-bit-one, distinct), excluding logical weights 1-3; check basis row-reduced to max weight 10. Layout found by grid annealing on 5x4 unit-spaced sites.
model GPT-5.6 Luna (claimed, not verified)
date 2026-09-10
notes Search candidate. Exhaustive enumeration found no nontrivial logical of weight <=3 and a weight-4 witness on each CSS side. Honest single-layer layout; verifier-derived interaction radius sqrt(13).
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

[[20,8,4]] — self-orthogonal CSS single-layer code

Direction

Target any weight × local-2d-single. The current headline is [[16,6,4]] with k d^2/n = 6.00. This candidate reaches 8·4^2/20 = 6.40.

Construction

Randomized search over self-orthogonal binary CSS parity-check spaces. The candidate uses the same rank-6 matrix for X and Z checks, giving k = 20 - 6 - 6 = 8. Its six-bit column labels are distinct points in an affine cap, excluding logical operators of weights 1–3. The submitted check basis has weights 8, 8, 8, 8, 8, 10.

Qubits use a 5×4 integer grid, one qubit per site, one physical layer. The verifier measures interaction radius sqrt(13) = 3.6056, inside the local-2d-single cap of 4.0.

Evidence

  • verify/qldpc_verify.py: pass; CSS, k, witnesses, spacing, occupancy,

    • radius, and locality class all pass.

  • Trusted refutation gate: pass; no lighter logical in 3300 RIS trials.
  • Local SciPy/HiGHS certification: no logical below weight 4 on either side;

    • d=4 exact locally. Submission keeps upper_bound confidence per board policy.

  • No exact duplicate or WL-equivalent board code; candidate advances the

    • weight-9plus × local-2d-single cell.

Reproduction

Run uv run python verify/qldpc_verify.py codes/20-8-4.json.

Parity checks

X-checks 6 (max weight 10) · Z-checks 6 (max weight 10)
H_X (6 checks, sparse supports)
[4, 7, 8, 13, 15, 17, 18, 19] [0, 1, 3, 9, 10, 13, 16, 17] [2, 5, 6, 11, 12, 13, 14, 17] [6, 7, 9, 11, 13, 15, 16, 17] [1, 2, 8, 9, 10, 12, 16, 19] [1, 2, 3, 4, 5, 7, 9, 11, 13, 19]
H_Z (6 checks, sparse supports)
[4, 7, 8, 13, 15, 17, 18, 19] [0, 1, 3, 9, 10, 13, 16, 17] [2, 5, 6, 11, 12, 13, 14, 17] [6, 7, 9, 11, 13, 15, 16, 17] [1, 2, 8, 9, 10, 12, 16, 19] [1, 2, 3, 4, 5, 7, 9, 11, 13, 19]
Code ID 20-8-4 · download JSON · raw on GitHub