[[54,16,6]] · QEC Challenge

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[[54,16,6]] d =

kd²/n

10.667

X/Z

0.0078

6.0828

layers

swaps

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 18, w_Z = 18 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[4, 6, 13, 15, 22, 24]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[27, 30, 36, 39, 45, 48]

certificate exact, d = 6 · CryptoMiniSat 5.14.7 SAT

X: no logical < 6 exists; Z: no logical < 6 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 18 · H_Z 18

qubit degrees H_X 9 · H_Z 9

trapping sets H_X (1,9)×54 (2,4)×54 (3,9)×3996 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,9): 54 (2,4): 54 (2,6): 162 (2,8): 135 (2,10): 135 (2,12): 324 (2,14): 189 (2,16): 216 (3,9): 3996 (3,11): 5967 (3,13): 5238 (3,15): 4401 (3,17): 1890 (3,19): 1350 (3,21): 270 (3,23): 108

trapping sets H_Z (1,9)×54 (2,4)×54 (3,9)×3996 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,9): 54 (2,4): 54 (2,6): 162 (2,8): 135 (2,10): 135 (2,12): 324 (2,14): 189 (2,16): 216 (3,9): 3996 (3,11): 5967 (3,13): 5238 (3,15): 4401 (3,17): 1890 (3,19): 1350 (3,21): 270 (3,23): 108

witness diameter X 6.3246 · Z 5.831 (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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (28)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 61 nearest-neighbor SWAPs per round in total, at most 4 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 Liangdong Lu and Guanmin Guo and Yang Liu and Ruipan Yang and @mathysrennela

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction Cyclic bicycle over F_2[Z_27], with circulant supports a=[0, 1, 2, 3, 4, 5, 6, 8, 25] and b=[0, 1, 3, 5, 6, 8, 11, 22, 25], from Table III of arXiv:2608.09115v1.

model GPT-5.6 Luna (claimed, not verified)

builds on arXiv:2608.09115

date 2026-08-10

notes The paper reports exact d=6. This staged record keeps the repository's witness-backed distance evidence as upper_bound until trusted certification.

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

[[54,16,6]] — cyclic bicycle reconstruction from arXiv:2608.09115v1

Direction & hypothesis

This is a direct reconstruction of the explicit cyclic bicycle row in Table III of arXiv:2608.09115v1. The paper reports [[54,16,6]] with exact distance 6. The reconstructed code advances the unrestricted weight-capable board cell.

What was searched

No new search was performed. Over F_2[Z_27], use H_X=[A|B] and H_Z=[B^T|A^T] with supp(a)={0,1,2,3,4,5,6,8,25} and supp(b)={0,1,3,5,6,8,11,22,25}.

Evidence trail

The matrices recompute to n=54, k=16, CSS commutation, and maximum check weight 18. The submission contains explicit witnesses produced by the repository builder. verify/validate_candidate.py returned passed: true, with no exact or WL-equivalent duplicate and board_advancing: true. The repository record uses witness-backed upper_bound confidence; the paper's exact distance claim is not treated as repository exact certification.

Dead ends

No additional search or confirmation was spent on dominated neighboring rows.

Tools

Model: GPT-5.6 Luna. Repository tooling: research/kit/bb.py, research/kit/css.py, research/kit/submit.py, and verify/validate_candidate.py. Author: @mathysrennela.

Reproduction

Use the support sets above with the cyclic two-block bicycle construction over Z_27. Source: arXiv:2608.09115v1, Table III.

Parity checks

X-checks 27 (max weight 18) · Z-checks 27 (max weight 18)

H_X (27 checks, sparse supports)

[0, 1, 2, 3, 4, 5, 6, 8, 25, 27, 28, 30, 32, 33, 35, 38, 49, 52] [1, 2, 3, 4, 5, 6, 7, 9, 26, 28, 29, 31, 33, 34, 36, 39, 50, 53] [0, 2, 3, 4, 5, 6, 7, 8, 10, 27, 29, 30, 32, 34, 35, 37, 40, 51] [1, 3, 4, 5, 6, 7, 8, 9, 11, 28, 30, 31, 33, 35, 36, 38, 41, 52] [2, 4, 5, 6, 7, 8, 9, 10, 12, 29, 31, 32, 34, 36, 37, 39, 42, 53] [3, 5, 6, 7, 8, 9, 10, 11, 13, 27, 30, 32, 33, 35, 37, 38, 40, 43] [4, 6, 7, 8, 9, 10, 11, 12, 14, 28, 31, 33, 34, 36, 38, 39, 41, 44] [5, 7, 8, 9, 10, 11, 12, 13, 15, 29, 32, 34, 35, 37, 39, 40, 42, 45] [6, 8, 9, 10, 11, 12, 13, 14, 16, 30, 33, 35, 36, 38, 40, 41, 43, 46] [7, 9, 10, 11, 12, 13, 14, 15, 17, 31, 34, 36, 37, 39, 41, 42, 44, 47] [8, 10, 11, 12, 13, 14, 15, 16, 18, 32, 35, 37, 38, 40, 42, 43, 45, 48] [9, 11, 12, 13, 14, 15, 16, 17, 19, 33, 36, 38, 39, 41, 43, 44, 46, 49] [10, 12, 13, 14, 15, 16, 17, 18, 20, 34, 37, 39, 40, 42, 44, 45, 47, 50] [11, 13, 14, 15, 16, 17, 18, 19, 21, 35, 38, 40, 41, 43, 45, 46, 48, 51] [12, 14, 15, 16, 17, 18, 19, 20, 22, 36, 39, 41, 42, 44, 46, 47, 49, 52] [13, 15, 16, 17, 18, 19, 20, 21, 23, 37, 40, 42, 43, 45, 47, 48, 50, 53] [14, 16, 17, 18, 19, 20, 21, 22, 24, 27, 38, 41, 43, 44, 46, 48, 49, 51] [15, 17, 18, 19, 20, 21, 22, 23, 25, 28, 39, 42, 44, 45, 47, 49, 50, 52] [16, 18, 19, 20, 21, 22, 23, 24, 26, 29, 40, 43, 45, 46, 48, 50, 51, 53] [0, 17, 19, 20, 21, 22, 23, 24, 25, 27, 30, 41, 44, 46, 47, 49, 51, 52] [1, 18, 20, 21, 22, 23, 24, 25, 26, 28, 31, 42, 45, 47, 48, 50, 52, 53] [0, 2, 19, 21, 22, 23, 24, 25, 26, 27, 29, 32, 43, 46, 48, 49, 51, 53] [0, 1, 3, 20, 22, 23, 24, 25, 26, 27, 28, 30, 33, 44, 47, 49, 50, 52] [0, 1, 2, 4, 21, 23, 24, 25, 26, 28, 29, 31, 34, 45, 48, 50, 51, 53] [0, 1, 2, 3, 5, 22, 24, 25, 26, 27, 29, 30, 32, 35, 46, 49, 51, 52] [0, 1, 2, 3, 4, 6, 23, 25, 26, 28, 30, 31, 33, 36, 47, 50, 52, 53] [0, 1, 2, 3, 4, 5, 7, 24, 26, 27, 29, 31, 32, 34, 37, 48, 51, 53]

H_Z (27 checks, sparse supports)

[0, 2, 5, 16, 19, 21, 22, 24, 26, 27, 29, 46, 48, 49, 50, 51, 52, 53] [0, 1, 3, 6, 17, 20, 22, 23, 25, 27, 28, 30, 47, 49, 50, 51, 52, 53] [1, 2, 4, 7, 18, 21, 23, 24, 26, 27, 28, 29, 31, 48, 50, 51, 52, 53] [0, 2, 3, 5, 8, 19, 22, 24, 25, 27, 28, 29, 30, 32, 49, 51, 52, 53] [1, 3, 4, 6, 9, 20, 23, 25, 26, 27, 28, 29, 30, 31, 33, 50, 52, 53] [0, 2, 4, 5, 7, 10, 21, 24, 26, 27, 28, 29, 30, 31, 32, 34, 51, 53] [0, 1, 3, 5, 6, 8, 11, 22, 25, 27, 28, 29, 30, 31, 32, 33, 35, 52] [1, 2, 4, 6, 7, 9, 12, 23, 26, 28, 29, 30, 31, 32, 33, 34, 36, 53] [0, 2, 3, 5, 7, 8, 10, 13, 24, 27, 29, 30, 31, 32, 33, 34, 35, 37] [1, 3, 4, 6, 8, 9, 11, 14, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38] [2, 4, 5, 7, 9, 10, 12, 15, 26, 29, 31, 32, 33, 34, 35, 36, 37, 39] [0, 3, 5, 6, 8, 10, 11, 13, 16, 30, 32, 33, 34, 35, 36, 37, 38, 40] [1, 4, 6, 7, 9, 11, 12, 14, 17, 31, 33, 34, 35, 36, 37, 38, 39, 41] [2, 5, 7, 8, 10, 12, 13, 15, 18, 32, 34, 35, 36, 37, 38, 39, 40, 42] [3, 6, 8, 9, 11, 13, 14, 16, 19, 33, 35, 36, 37, 38, 39, 40, 41, 43] [4, 7, 9, 10, 12, 14, 15, 17, 20, 34, 36, 37, 38, 39, 40, 41, 42, 44] [5, 8, 10, 11, 13, 15, 16, 18, 21, 35, 37, 38, 39, 40, 41, 42, 43, 45] [6, 9, 11, 12, 14, 16, 17, 19, 22, 36, 38, 39, 40, 41, 42, 43, 44, 46] [7, 10, 12, 13, 15, 17, 18, 20, 23, 37, 39, 40, 41, 42, 43, 44, 45, 47] [8, 11, 13, 14, 16, 18, 19, 21, 24, 38, 40, 41, 42, 43, 44, 45, 46, 48] [9, 12, 14, 15, 17, 19, 20, 22, 25, 39, 41, 42, 43, 44, 45, 46, 47, 49] [10, 13, 15, 16, 18, 20, 21, 23, 26, 40, 42, 43, 44, 45, 46, 47, 48, 50] [0, 11, 14, 16, 17, 19, 21, 22, 24, 41, 43, 44, 45, 46, 47, 48, 49, 51] [1, 12, 15, 17, 18, 20, 22, 23, 25, 42, 44, 45, 46, 47, 48, 49, 50, 52] [2, 13, 16, 18, 19, 21, 23, 24, 26, 43, 45, 46, 47, 48, 49, 50, 51, 53] [0, 3, 14, 17, 19, 20, 22, 24, 25, 27, 44, 46, 47, 48, 49, 50, 51, 52] [1, 4, 15, 18, 20, 21, 23, 25, 26, 28, 45, 47, 48, 49, 50, 51, 52, 53]

Code ID 54-16-6 · download JSON · raw on GitHub