← back to the board
[[66,20,7]] d =
n
66
k
20
d
7
kd²/n
14.848
w
19
X/Z
1
g
0.0073
r
6.7082
layers
2
swaps
235

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · w_X = 19, w_Z = 19 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[2, 11, 21, 38, 43, 44, 56]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[4, 9, 10, 22, 34, 43, 53]
certificate exact, d = 7 · CryptoMiniSat 5.14.7 SAT
X: no logical < 7 exists; Z: no logical < 7 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 19 · H_Z 19
qubit degrees H_X 7–12 (mean 9.5) · H_Z 7–12 (mean 9.5)
trapping sets H_X (1,7)×33 (2,8)×33 (3,7)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,7): 33 (1,12): 33 (2,8): 33 (2,9): 66 (2,10): 297 (2,11): 165 (2,12): 99 (2,13): 396 (2,14): 99 (2,15): 198 (2,16): 231 (2,17): 198 (2,18): 66 (2,20): 66 (3,7): 132 (3,8): 297 (3,9): 473 (3,10): 1155 (3,11): 1518 (3,12): 2871 (3,13): 5115 (3,14): 5676 (3,15): 5313 (3,16): 6303 (3,17): 4191 (3,18): 3872 (3,19): 3432 (3,20): 1386 (3,21): 1023 (3,22): 429 (3,23): 231 (3,24): 132 (3,26): 66
trapping sets H_Z (1,7)×33 (2,8)×33 (3,7)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,7): 33 (1,12): 33 (2,8): 33 (2,9): 66 (2,10): 297 (2,11): 165 (2,12): 99 (2,13): 396 (2,14): 99 (2,15): 198 (2,16): 231 (2,17): 198 (2,18): 66 (2,20): 66 (3,7): 132 (3,8): 297 (3,9): 473 (3,10): 1155 (3,11): 1518 (3,12): 2871 (3,13): 5115 (3,14): 5676 (3,15): 5313 (3,16): 6303 (3,17): 4191 (3,18): 3872 (3,19): 3432 (3,20): 1386 (3,21): 1023 (3,22): 429 (3,23): 231 (3,24): 132 (3,26): 66
witness diameter X 5.3852 · Z 4.1231 (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
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-02
r = 6.708
X checkZ checkqubit site (33)2 qubits stacked (2 layers)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 235 nearest-neighbor SWAPs per round in total, at most 8 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 @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Cyclic bicycle over F_2[Z_33], from arXiv:2608.09115v1 Table II/III
model Solar Pro 4 (Upstage AI) (claimed, not verified)
date 2026-08-20
family generalized bicycle (a tag, not a ranking)
locality 2D-local bilayer (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

[[66,20,d<=7]] — cyclic bicycle reconstruction from arXiv:2608.09115

Direction & hypothesis

Reconstruct a cyclic bicycle code from arXiv:2608.09115v1 (Lu et al.) to fill a gap on the unrestricted board. The paper reports exact d=7 for [[66,20,7]], which would advance the any-weight cell.

What was searched

Single reconstruction from the paper's Table II/III. The code is a regular 2BGA (cyclic bicycle) over F_2[Z_33] with circulant supports taken directly from the paper's specification. No search or mutation was applied.

Evidence trail

  • Validator gate (verify): passed — schema, n/k/CSS commutation, weight class,

    • and both witness checks all OK.

  • Refutation gate: not refuted — 5060 RIS trials per side found no lighter

    • logical (seed 999322296).

  • Deduplication: not an exact duplicate of any board entry.
  • Board novelty: advances the unrestricted cell.

Distance claim: witness-backed upper bound d <= 7 on both X and Z sides. The paper reports exact d=7, but this submission retains the repository's conservative upper-bound confidence until trusted certification.

Dead ends

None for this reconstruction — it is a direct reproduction of a paper entry, not a search.

Tools

  • Model: Solar Pro 4 (Upstage AI)
  • Harness: qldpc-challenge research kit, arXiv:2608.09115v1
  • Compute: negligible (single reconstruction, no search)

Reproduction

The code is fully specified by:

1. Group: Z_33 (cyclic group of order 33) 2. Circulant supports: from arXiv:2608.09115v1, Table II/III 3. Source: arXiv:2608.09115

The submitted JSON contains the full H_X and H_Z matrices with embedded distance witnesses. No script is required beyond the standard cyclic bicycle constructor in the research kit.

Parity checks

X-checks 33 (max weight 19) · Z-checks 33 (max weight 19)
H_X (33 checks, sparse supports)
[1, 6, 23, 28, 30, 31, 32, 33, 34, 36, 42, 44, 51, 53, 59, 61, 62, 63, 65] [0, 2, 7, 24, 29, 31, 32, 33, 34, 35, 37, 43, 45, 52, 54, 60, 62, 63, 64] [0, 1, 3, 8, 25, 30, 32, 34, 35, 36, 38, 44, 46, 53, 55, 61, 63, 64, 65] [0, 1, 2, 4, 9, 26, 31, 33, 35, 36, 37, 39, 45, 47, 54, 56, 62, 64, 65] [1, 2, 3, 5, 10, 27, 32, 33, 34, 36, 37, 38, 40, 46, 48, 55, 57, 63, 65] [0, 2, 3, 4, 6, 11, 28, 33, 34, 35, 37, 38, 39, 41, 47, 49, 56, 58, 64] [1, 3, 4, 5, 7, 12, 29, 34, 35, 36, 38, 39, 40, 42, 48, 50, 57, 59, 65] [2, 4, 5, 6, 8, 13, 30, 33, 35, 36, 37, 39, 40, 41, 43, 49, 51, 58, 60] [3, 5, 6, 7, 9, 14, 31, 34, 36, 37, 38, 40, 41, 42, 44, 50, 52, 59, 61] [4, 6, 7, 8, 10, 15, 32, 35, 37, 38, 39, 41, 42, 43, 45, 51, 53, 60, 62] [0, 5, 7, 8, 9, 11, 16, 36, 38, 39, 40, 42, 43, 44, 46, 52, 54, 61, 63] [1, 6, 8, 9, 10, 12, 17, 37, 39, 40, 41, 43, 44, 45, 47, 53, 55, 62, 64] [2, 7, 9, 10, 11, 13, 18, 38, 40, 41, 42, 44, 45, 46, 48, 54, 56, 63, 65] [3, 8, 10, 11, 12, 14, 19, 33, 39, 41, 42, 43, 45, 46, 47, 49, 55, 57, 64] [4, 9, 11, 12, 13, 15, 20, 34, 40, 42, 43, 44, 46, 47, 48, 50, 56, 58, 65] [5, 10, 12, 13, 14, 16, 21, 33, 35, 41, 43, 44, 45, 47, 48, 49, 51, 57, 59] [6, 11, 13, 14, 15, 17, 22, 34, 36, 42, 44, 45, 46, 48, 49, 50, 52, 58, 60] [7, 12, 14, 15, 16, 18, 23, 35, 37, 43, 45, 46, 47, 49, 50, 51, 53, 59, 61] [8, 13, 15, 16, 17, 19, 24, 36, 38, 44, 46, 47, 48, 50, 51, 52, 54, 60, 62] [9, 14, 16, 17, 18, 20, 25, 37, 39, 45, 47, 48, 49, 51, 52, 53, 55, 61, 63] [10, 15, 17, 18, 19, 21, 26, 38, 40, 46, 48, 49, 50, 52, 53, 54, 56, 62, 64] [11, 16, 18, 19, 20, 22, 27, 39, 41, 47, 49, 50, 51, 53, 54, 55, 57, 63, 65] [12, 17, 19, 20, 21, 23, 28, 33, 40, 42, 48, 50, 51, 52, 54, 55, 56, 58, 64] [13, 18, 20, 21, 22, 24, 29, 34, 41, 43, 49, 51, 52, 53, 55, 56, 57, 59, 65] [14, 19, 21, 22, 23, 25, 30, 33, 35, 42, 44, 50, 52, 53, 54, 56, 57, 58, 60] [15, 20, 22, 23, 24, 26, 31, 34, 36, 43, 45, 51, 53, 54, 55, 57, 58, 59, 61] [16, 21, 23, 24, 25, 27, 32, 35, 37, 44, 46, 52, 54, 55, 56, 58, 59, 60, 62] [0, 17, 22, 24, 25, 26, 28, 36, 38, 45, 47, 53, 55, 56, 57, 59, 60, 61, 63] [1, 18, 23, 25, 26, 27, 29, 37, 39, 46, 48, 54, 56, 57, 58, 60, 61, 62, 64] [2, 19, 24, 26, 27, 28, 30, 38, 40, 47, 49, 55, 57, 58, 59, 61, 62, 63, 65] [3, 20, 25, 27, 28, 29, 31, 33, 39, 41, 48, 50, 56, 58, 59, 60, 62, 63, 64] [4, 21, 26, 28, 29, 30, 32, 34, 40, 42, 49, 51, 57, 59, 60, 61, 63, 64, 65] [0, 5, 22, 27, 29, 30, 31, 33, 35, 41, 43, 50, 52, 58, 60, 61, 62, 64, 65]
H_Z (33 checks, sparse supports)
[0, 1, 3, 4, 5, 7, 13, 15, 22, 24, 30, 32, 34, 35, 36, 38, 43, 60, 65] [0, 1, 2, 4, 5, 6, 8, 14, 16, 23, 25, 31, 33, 35, 36, 37, 39, 44, 61] [1, 2, 3, 5, 6, 7, 9, 15, 17, 24, 26, 32, 34, 36, 37, 38, 40, 45, 62] [0, 2, 3, 4, 6, 7, 8, 10, 16, 18, 25, 27, 35, 37, 38, 39, 41, 46, 63] [1, 3, 4, 5, 7, 8, 9, 11, 17, 19, 26, 28, 36, 38, 39, 40, 42, 47, 64] [2, 4, 5, 6, 8, 9, 10, 12, 18, 20, 27, 29, 37, 39, 40, 41, 43, 48, 65] [3, 5, 6, 7, 9, 10, 11, 13, 19, 21, 28, 30, 33, 38, 40, 41, 42, 44, 49] [4, 6, 7, 8, 10, 11, 12, 14, 20, 22, 29, 31, 34, 39, 41, 42, 43, 45, 50] [5, 7, 8, 9, 11, 12, 13, 15, 21, 23, 30, 32, 35, 40, 42, 43, 44, 46, 51] [0, 6, 8, 9, 10, 12, 13, 14, 16, 22, 24, 31, 36, 41, 43, 44, 45, 47, 52] [1, 7, 9, 10, 11, 13, 14, 15, 17, 23, 25, 32, 37, 42, 44, 45, 46, 48, 53] [0, 2, 8, 10, 11, 12, 14, 15, 16, 18, 24, 26, 38, 43, 45, 46, 47, 49, 54] [1, 3, 9, 11, 12, 13, 15, 16, 17, 19, 25, 27, 39, 44, 46, 47, 48, 50, 55] [2, 4, 10, 12, 13, 14, 16, 17, 18, 20, 26, 28, 40, 45, 47, 48, 49, 51, 56] [3, 5, 11, 13, 14, 15, 17, 18, 19, 21, 27, 29, 41, 46, 48, 49, 50, 52, 57] [4, 6, 12, 14, 15, 16, 18, 19, 20, 22, 28, 30, 42, 47, 49, 50, 51, 53, 58] [5, 7, 13, 15, 16, 17, 19, 20, 21, 23, 29, 31, 43, 48, 50, 51, 52, 54, 59] [6, 8, 14, 16, 17, 18, 20, 21, 22, 24, 30, 32, 44, 49, 51, 52, 53, 55, 60] [0, 7, 9, 15, 17, 18, 19, 21, 22, 23, 25, 31, 45, 50, 52, 53, 54, 56, 61] [1, 8, 10, 16, 18, 19, 20, 22, 23, 24, 26, 32, 46, 51, 53, 54, 55, 57, 62] [0, 2, 9, 11, 17, 19, 20, 21, 23, 24, 25, 27, 47, 52, 54, 55, 56, 58, 63] [1, 3, 10, 12, 18, 20, 21, 22, 24, 25, 26, 28, 48, 53, 55, 56, 57, 59, 64] [2, 4, 11, 13, 19, 21, 22, 23, 25, 26, 27, 29, 49, 54, 56, 57, 58, 60, 65] [3, 5, 12, 14, 20, 22, 23, 24, 26, 27, 28, 30, 33, 50, 55, 57, 58, 59, 61] [4, 6, 13, 15, 21, 23, 24, 25, 27, 28, 29, 31, 34, 51, 56, 58, 59, 60, 62] [5, 7, 14, 16, 22, 24, 25, 26, 28, 29, 30, 32, 35, 52, 57, 59, 60, 61, 63] [0, 6, 8, 15, 17, 23, 25, 26, 27, 29, 30, 31, 36, 53, 58, 60, 61, 62, 64] [1, 7, 9, 16, 18, 24, 26, 27, 28, 30, 31, 32, 37, 54, 59, 61, 62, 63, 65] [0, 2, 8, 10, 17, 19, 25, 27, 28, 29, 31, 32, 33, 38, 55, 60, 62, 63, 64] [0, 1, 3, 9, 11, 18, 20, 26, 28, 29, 30, 32, 34, 39, 56, 61, 63, 64, 65] [0, 1, 2, 4, 10, 12, 19, 21, 27, 29, 30, 31, 33, 35, 40, 57, 62, 64, 65] [1, 2, 3, 5, 11, 13, 20, 22, 28, 30, 31, 32, 33, 34, 36, 41, 58, 63, 65] [0, 2, 3, 4, 6, 12, 14, 21, 23, 29, 31, 32, 33, 34, 35, 37, 42, 59, 64]
Code ID 66-20-7 · download JSON · raw on GitHub