← back to the board
[[84,14,10]] d =
n
84
k
14
d
10
kd²/n
16.667
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[2, 7, 11, 18, 38, 51, 52, 60, 65, 67]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[15, 22, 31, 33, 40, 42, 44, 49, 58, 75]
certificate exact, d = 10 · CryptoMiniSat 5.14 SAT
X: no logical < 10 exists; Z: no logical < 10 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 · H_Z 8
qubit degrees H_X 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×42 (2,4)×168 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 42 (1,5): 42 (2,4): 168 (2,6): 546 (2,8): 420 (3,3): 42 (3,5): 1008 (3,7): 6510 (3,9): 9758 (3,11): 4200 (3,13): 420
trapping sets H_Z (1,3)×42 (2,4)×168 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 42 (1,5): 42 (2,4): 168 (2,6): 546 (2,8): 420 (3,3): 42 (3,5): 1008 (3,7): 6510 (3,9): 9758 (3,11): 4200 (3,13): 420

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction 2BGA reconstruction from Lin-Pryadko catalogue github.com/QEC-pages/2BGA-codes @ 403d194c, SmallGroup(42,3), a=1+g8+g27, b=1+g3+g16+g39+g42
builds on arXiv:2306.16400
date 2026-09-17
notes Reconstruction of a published Lin-Pryadko 2BGA code (SmallGroup(42,3), [[84,14,10]]). Checked against the current board: not an exact or WL-equivalent duplicate of any existing entry.
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

[[84,14,10]] — published 2BGA code on SmallGroup(42,3)

The distance is a witness-backed upper bound d <= 10, not an exact certificate.

Direction & hypothesis

Reconstruct a published code absent from the unrestricted weight-8 frontier for hackathon issue 1155. Against board commit 6995fc4f72a33fcbea27a244da230fcfc370e108, the trusted candidate gate labels this candidate board-advancing. Eligibility dimensions are n=84, w=8 and claimed d=10. No layout is supplied, so this targets the weight <= 8 prize, not the weight <= 6 or locality prizes.

What was searched

Parsed 72,637 rows from both archives in github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. Used published n, k, d and filename weight for a parameter-only shortlist against 663 board entries. Preliminary counts: 64,684 dominated rows, 184 tied rows and 7,769 potential gap rows, collapsing to 1,322 nondominated-or-tied parameter points. These are not validated finds: archive row conventions and all reported distances have not been independently checked across the catalogue.

Reconstructed only the first shortlisted row, then stopped at the first gate-passing board advance. Source: that pinned repository's nonabelian.zip, member diswtnonabelian_wt8wtL3_order42_k14.txt, line 6:

42 3 14 10 [ 8, 27 ] [ 3, 16, 39, 42 ]

Evidence trail

  • GAP reconstruction and the research kit's 2BGA constructor reproduce n=84,

    • k=14, maximum check weight 8 and CSS commutation. Combined X/Z check incidence connects all 84 qubits; no direct sum is used.

  • The submission's own witness search found weight-10 X and Z logicals.
  • Independent BP+OSD estimator: weight-10 logical witnesses on both sides,

    • verdict corroborated.

  • Trusted validate_candidate with refutation enabled: passed=true; no failed

    • structural checks; no lighter logical found; no exact duplicate or WL-equivalent entry; board_advancing=true, dominated_by empty.

Screening efficiency k*d^2/n = 16.6667, based on the witnessed upper bound. No exact-distance certification was attempted. This construction is explicitly known literature: it is a reconstruction, not a discovery.

Dead ends

The preceding random bivariate-bicycle pilot produced only a dominated [[72,8,6]] candidate; catalogue lookup avoided another blind sweep. The second and third catalogue shortlist rows were not reconstructed once this candidate passed. Some archive rows explicitly include the identity whereas the selected row excludes it; do not apply one support-normalization convention blindly to the full catalogue. This row's indexing and weight were confirmed by reconstruction, not assumed from the filename.

Tools

Union Alpha (maker currently anonymous), Zed; GAP SmallGroups, the research kit's group_algebra.build_2bga, CSS rank functions, the submission packager, an existing BP+OSD estimator and the unchanged trusted verifier. Reconstruction, packaging, decoder and first candidate gate took approximately 28 seconds locally. No remote compute or trusted-stack modifications.

Reproduction

Use G=SmallGroup(42,3) in GAP and e=Elements(G), with 1-based indexing. Set a=1+e[8]+e[27] and b=1+e[3]+e[16]+e[39]+e[42]. Export the multiplication table in that exact element order; convert indices to zero-based for the research kit. Build H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T]. Package with confidence upper_bound. The construction is completely specified by the pinned source and these supports.

Parity checks

X-checks 42 (max weight 8) · Z-checks 42 (max weight 8)
H_X (42 checks, sparse supports)
[0, 25, 40, 42, 52, 59, 63, 72] [1, 35, 38, 43, 49, 56, 60, 75] [2, 8, 31, 42, 44, 58, 65, 69] [3, 30, 32, 45, 46, 52, 77, 78] [4, 5, 39, 43, 46, 55, 62, 66] [5, 21, 41, 44, 47, 49, 74, 80] [6, 14, 36, 44, 48, 64, 71, 75] [0, 7, 37, 45, 49, 51, 58, 81] [8, 18, 36, 46, 50, 59, 70, 82] [9, 10, 41, 46, 51, 61, 68, 72] [10, 11, 27, 47, 48, 52, 55, 79] [11, 28, 33, 44, 53, 56, 67, 83] [3, 12, 20, 43, 48, 54, 70, 77] [2, 13, 40, 49, 55, 57, 64, 83] [3, 14, 24, 50, 51, 56, 65, 76] [11, 15, 16, 42, 51, 57, 67, 74] [16, 17, 33, 52, 54, 58, 61, 82] [1, 17, 34, 48, 53, 59, 62, 73] [7, 18, 26, 46, 54, 60, 76, 81] [6, 8, 19, 53, 55, 61, 63, 70] [7, 20, 30, 56, 57, 62, 71, 80] [17, 21, 22, 44, 57, 63, 73, 79] [1, 22, 23, 50, 58, 60, 64, 67] [4, 23, 38, 54, 59, 65, 68, 78] [13, 24, 32, 51, 60, 66, 80, 83] [12, 14, 25, 59, 61, 67, 69, 76] [0, 13, 26, 47, 62, 63, 68, 77] [23, 27, 28, 48, 63, 69, 78, 82] [4, 28, 29, 56, 64, 66, 70, 73] [5, 9, 29, 45, 60, 65, 71, 74] [19, 30, 37, 47, 53, 57, 66, 72] [18, 20, 31, 65, 67, 73, 75, 80] [2, 19, 32, 52, 68, 69, 74, 81] [29, 33, 34, 45, 50, 54, 69, 75] [9, 34, 35, 62, 70, 72, 76, 78] [10, 15, 35, 49, 66, 71, 77, 79] [24, 26, 36, 43, 47, 71, 73, 78] [6, 25, 37, 58, 74, 75, 79, 83] [15, 38, 39, 42, 45, 68, 76, 80] [16, 21, 39, 55, 72, 77, 81, 82] [12, 31, 40, 43, 53, 64, 79, 82] [22, 27, 41, 42, 50, 61, 81, 83]
H_Z (42 checks, sparse supports)
[0, 2, 15, 38, 41, 42, 49, 68] [1, 4, 12, 36, 40, 43, 59, 64] [2, 5, 6, 11, 21, 44, 55, 74] [3, 7, 29, 33, 38, 45, 54, 56] [3, 4, 8, 9, 18, 46, 65, 70] [5, 10, 26, 30, 36, 46, 47, 71] [6, 10, 12, 17, 27, 48, 61, 79] [1, 5, 7, 13, 35, 49, 60, 62] [8, 14, 22, 33, 41, 44, 50, 61] [7, 9, 14, 15, 24, 51, 71, 76] [0, 3, 10, 16, 32, 51, 52, 77] [11, 17, 19, 30, 40, 52, 53, 57] [12, 16, 18, 23, 33, 54, 67, 82] [4, 10, 13, 19, 39, 55, 66, 68] [1, 11, 14, 20, 28, 48, 56, 67] [13, 15, 20, 21, 30, 57, 77, 80] [2, 7, 16, 22, 37, 57, 58, 81] [0, 8, 17, 23, 25, 58, 59, 63] [1, 18, 22, 24, 29, 50, 60, 73] [9, 16, 19, 25, 41, 61, 72, 74] [4, 17, 20, 26, 34, 54, 62, 73] [0, 19, 21, 26, 27, 47, 63, 81] [6, 13, 22, 28, 40, 63, 64, 83] [2, 14, 23, 29, 31, 64, 65, 69] [4, 24, 28, 30, 35, 56, 66, 78] [11, 15, 22, 25, 31, 42, 67, 79] [9, 23, 26, 32, 38, 60, 68, 78] [2, 25, 27, 32, 33, 52, 69, 83] [8, 12, 19, 28, 34, 53, 69, 70] [6, 20, 29, 35, 36, 70, 71, 75] [0, 9, 30, 34, 39, 45, 62, 72] [17, 21, 28, 31, 36, 44, 73, 82] [5, 15, 29, 32, 37, 45, 66, 74] [1, 6, 31, 33, 37, 53, 58, 75] [14, 18, 25, 34, 38, 59, 75, 76] [3, 12, 26, 35, 39, 43, 76, 77] [3, 23, 27, 34, 36, 48, 50, 78] [10, 21, 35, 37, 40, 49, 72, 79] [5, 20, 24, 31, 38, 43, 65, 80] [7, 18, 32, 39, 41, 46, 80, 81] [8, 16, 27, 39, 40, 42, 55, 82] [11, 13, 24, 37, 41, 47, 51, 83]
Code ID 84-14-10 · download JSON · raw on GitHub