[[42,16,6]] · QEC Challenge

[[42,16,6]] d =

kd²/n

13.714

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 22, w_Z = 22 (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, 8, 10, 17, 21, 33]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[3, 14, 18, 32, 35, 39]

certificate exact, d = 6 · CryptoMiniSat 5.14 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 22 · H_Z 22

qubit degrees H_X 9–13 (mean 11.0) · H_Z 9–13 (mean 11.0)

trapping sets H_X (1,9)×21 (2,6)×42 (3,5)×189 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,9): 21 (1,13): 21 (2,6): 42 (2,8): 126 (2,10): 273 (2,12): 336 (2,14): 63 (2,16): 21 (3,5): 189 (3,7): 1113 (3,9): 4151 (3,11): 2877 (3,13): 2520 (3,15): 546 (3,17): 84

trapping sets H_Z (1,9)×21 (2,6)×42 (3,5)×189 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,9): 21 (1,13): 21 (2,6): 42 (2,8): 126 (2,10): 273 (2,12): 336 (2,14): 63 (2,16): 21 (3,5): 189 (3,7): 1113 (3,9): 4151 (3,11): 2877 (3,13): 2520 (3,15): 546 (3,17): 84

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_21], with circulant supports a=[0, 1, 2, 8, 9, 11, 17, 18, 20] and b=[0, 1, 2, 3, 4, 6, 8, 11, 12, 14, 18, 19, 20], 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 unrestricted (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

[[42,16,6]] — 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 [[42,16,6]] 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_21], use H_X=[A|B] and H_Z=[B^T|A^T] with supp(a)={0,1,2,8,9,11,17,18,20} and supp(b)={0,1,2,3,4,6,8,11,12,14,18,19,20}.

Evidence trail

The matrices recompute to n=42, k=16, CSS commutation, and maximum check weight 22. 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

The nearby [[42,12,4]] row is already represented on the board and was not duplicated.

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_21. Source: arXiv:2608.09115v1, Table III.

Parity checks

X-checks 21 (max weight 22) · Z-checks 21 (max weight 22)

H_X (21 checks, sparse supports)

[0, 1, 2, 8, 9, 11, 17, 18, 20, 21, 22, 23, 24, 25, 27, 29, 32, 33, 35, 39, 40, 41] [0, 1, 2, 3, 9, 10, 12, 18, 19, 21, 22, 23, 24, 25, 26, 28, 30, 33, 34, 36, 40, 41] [1, 2, 3, 4, 10, 11, 13, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 34, 35, 37, 41] [0, 2, 3, 4, 5, 11, 12, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 35, 36, 38] [0, 1, 3, 4, 5, 6, 12, 13, 15, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 36, 37, 39] [1, 2, 4, 5, 6, 7, 13, 14, 16, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 37, 38, 40] [2, 3, 5, 6, 7, 8, 14, 15, 17, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 38, 39, 41] [3, 4, 6, 7, 8, 9, 15, 16, 18, 21, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 39, 40] [4, 5, 7, 8, 9, 10, 16, 17, 19, 22, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 40, 41] [5, 6, 8, 9, 10, 11, 17, 18, 20, 21, 23, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 41] [0, 6, 7, 9, 10, 11, 12, 18, 19, 21, 22, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 39] [1, 7, 8, 10, 11, 12, 13, 19, 20, 22, 23, 25, 29, 30, 31, 32, 33, 34, 35, 36, 38, 40] [0, 2, 8, 9, 11, 12, 13, 14, 20, 23, 24, 26, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41] [0, 1, 3, 9, 10, 12, 13, 14, 15, 21, 24, 25, 27, 31, 32, 33, 34, 35, 36, 37, 38, 40] [1, 2, 4, 10, 11, 13, 14, 15, 16, 22, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 39, 41] [2, 3, 5, 11, 12, 14, 15, 16, 17, 21, 23, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40] [3, 4, 6, 12, 13, 15, 16, 17, 18, 22, 24, 27, 28, 30, 34, 35, 36, 37, 38, 39, 40, 41] [4, 5, 7, 13, 14, 16, 17, 18, 19, 21, 23, 25, 28, 29, 31, 35, 36, 37, 38, 39, 40, 41] [5, 6, 8, 14, 15, 17, 18, 19, 20, 21, 22, 24, 26, 29, 30, 32, 36, 37, 38, 39, 40, 41] [0, 6, 7, 9, 15, 16, 18, 19, 20, 21, 22, 23, 25, 27, 30, 31, 33, 37, 38, 39, 40, 41] [0, 1, 7, 8, 10, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 31, 32, 34, 38, 39, 40, 41]

H_Z (21 checks, sparse supports)

[0, 1, 2, 3, 7, 9, 10, 13, 15, 17, 18, 19, 20, 21, 22, 24, 25, 31, 33, 34, 40, 41] [0, 1, 2, 3, 4, 8, 10, 11, 14, 16, 18, 19, 20, 21, 22, 23, 25, 26, 32, 34, 35, 41] [0, 1, 2, 3, 4, 5, 9, 11, 12, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 33, 35, 36] [0, 1, 2, 3, 4, 5, 6, 10, 12, 13, 16, 18, 20, 22, 23, 24, 25, 27, 28, 34, 36, 37] [0, 1, 2, 3, 4, 5, 6, 7, 11, 13, 14, 17, 19, 23, 24, 25, 26, 28, 29, 35, 37, 38] [1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 18, 20, 24, 25, 26, 27, 29, 30, 36, 38, 39] [0, 2, 3, 4, 5, 6, 7, 8, 9, 13, 15, 16, 19, 25, 26, 27, 28, 30, 31, 37, 39, 40] [1, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 17, 20, 26, 27, 28, 29, 31, 32, 38, 40, 41] [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 18, 21, 27, 28, 29, 30, 32, 33, 39, 41] [1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 21, 22, 28, 29, 30, 31, 33, 34, 40] [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 17, 19, 20, 22, 23, 29, 30, 31, 32, 34, 35, 41] [0, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 18, 20, 21, 23, 24, 30, 31, 32, 33, 35, 36] [0, 1, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 19, 22, 24, 25, 31, 32, 33, 34, 36, 37] [1, 2, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 20, 23, 25, 26, 32, 33, 34, 35, 37, 38] [0, 2, 3, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 24, 26, 27, 33, 34, 35, 36, 38, 39] [1, 3, 4, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 25, 27, 28, 34, 35, 36, 37, 39, 40] [2, 4, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 26, 28, 29, 35, 36, 37, 38, 40, 41] [3, 5, 6, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 27, 29, 30, 36, 37, 38, 39, 41] [0, 4, 6, 7, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 28, 30, 31, 37, 38, 39, 40] [0, 1, 5, 7, 8, 11, 13, 15, 16, 17, 18, 19, 20, 22, 23, 29, 31, 32, 38, 39, 40, 41] [0, 1, 2, 6, 8, 9, 12, 14, 16, 17, 18, 19, 20, 21, 23, 24, 30, 32, 33, 39, 40, 41]

Code ID 42-16-6 · download JSON · raw on GitHub