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[[100,4,14]] d ≤
n
100
k
4
d
14
kd²/n
7.84
w
8
X/Z
1
g
0.0033
r
7.0
layers
2
swaps
611

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[4, 10, 15, 21, 26, 30, 53, 57, 58, 59, 65, 68, 71, 76]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 3, 5, 13, 38, 47, 49, 51, 63, 81, 82, 91, 96, 98]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 4 · H_Z 4
trapping sets H_X (1,4)×100 (2,4)×150 (3,4)×300 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 100 (2,4): 150 (2,6): 1100 (3,4): 300 (3,6): 4200 (3,8): 16150 (3,10): 1300
trapping sets H_Z (1,4)×100 (2,4)×150 (3,4)×300 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 100 (2,4): 150 (2,6): 1100 (3,4): 300 (3,6): 4200 (3,8): 16150 (3,10): 1300
witness diameter X 8.0623 · Z 8.9443 (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 = 7
X checkZ checkqubit site (51)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 611 nearest-neighbor SWAPs per round in total, at most 12 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 2BGA (arXiv:2306.16400) on SmallGroup(50,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order50_k4.txt, line 14. nonidentity supports a=[2, 3, 6], b=[3, 8, 46].
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). Checked against the live board: not equivalent to any existing entry.
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

[[100,4,14]] — two-block group-algebra code on SmallGroup(50,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 7.84.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order50_k4.txt, line 14. This row is SmallGroup(50,3) with nonidentity GAP supports a=[2, 3, 6], b=[3, 8, 46]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 14, not an exact certificate. The published distance is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(50,3), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[2, 3, 6], b=[3, 8, 46] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 100 - rank(HX) - rank(HZ) = 4. Pass to the repository submission builder with confidence upper_bound; save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 50 (max weight 8) · Z-checks 50 (max weight 8)
H_X (50 checks, sparse supports)
[0, 1, 5, 20, 50, 63, 70, 98] [0, 1, 24, 25, 51, 67, 75, 99] [0, 2, 4, 10, 50, 52, 71, 74] [3, 5, 11, 30, 53, 70, 72, 80] [1, 2, 4, 33, 51, 54, 76, 79] [0, 3, 5, 34, 55, 75, 77, 84] [2, 6, 9, 17, 52, 56, 80, 83] [3, 7, 10, 18, 50, 53, 57, 81] [8, 11, 19, 38, 58, 80, 82, 88] [4, 6, 9, 40, 54, 59, 84, 87] [2, 5, 7, 10, 51, 55, 60, 85] [3, 8, 11, 41, 61, 84, 86, 91] [6, 12, 16, 26, 53, 56, 62, 90] [7, 13, 17, 27, 52, 57, 63, 88] [8, 14, 18, 28, 53, 58, 64, 89] [15, 19, 29, 44, 65, 88, 90, 94] [9, 12, 16, 45, 55, 59, 66, 93] [6, 10, 13, 17, 54, 60, 67, 91] [7, 11, 14, 18, 55, 61, 68, 92] [8, 15, 19, 46, 69, 91, 93, 96] [12, 20, 25, 34, 57, 62, 70, 95] [13, 21, 26, 35, 56, 58, 63, 71] [14, 22, 27, 36, 57, 64, 72, 94] [15, 23, 28, 37, 58, 65, 73, 95] [1, 24, 29, 48, 56, 74, 94, 98] [16, 20, 25, 48, 60, 66, 75, 97] [12, 17, 21, 26, 59, 61, 67, 76] [13, 18, 22, 27, 60, 68, 77, 96] [14, 19, 23, 28, 61, 69, 78, 97] [15, 24, 29, 49, 59, 79, 96, 99] [21, 30, 34, 41, 62, 64, 71, 80] [22, 31, 35, 42, 63, 65, 72, 81] [23, 32, 36, 43, 64, 73, 82, 98] [4, 24, 33, 37, 62, 65, 74, 83] [20, 26, 30, 34, 66, 68, 76, 84] [21, 27, 31, 35, 67, 69, 77, 85] [22, 28, 32, 36, 68, 78, 86, 99] [23, 29, 33, 37, 66, 69, 79, 87] [31, 38, 41, 46, 71, 73, 81, 88] [32, 39, 42, 47, 72, 74, 82, 89] [9, 33, 40, 43, 70, 73, 83, 90] [30, 35, 38, 41, 76, 78, 85, 91] [31, 36, 39, 42, 77, 79, 86, 92] [32, 37, 40, 43, 75, 78, 87, 93] [39, 44, 46, 49, 81, 83, 89, 94] [16, 40, 45, 47, 50, 82, 90, 95] [38, 42, 44, 46, 85, 87, 92, 96] [39, 43, 45, 47, 51, 86, 93, 97] [25, 45, 48, 49, 52, 89, 95, 98] [44, 47, 48, 49, 54, 92, 97, 99]
H_Z (50 checks, sparse supports)
[0, 2, 7, 45, 50, 51, 52, 55] [1, 4, 10, 47, 50, 51, 54, 74] [2, 6, 13, 48, 52, 54, 56, 60] [3, 7, 12, 14, 53, 55, 57, 61] [4, 9, 17, 49, 52, 54, 59, 83] [5, 10, 16, 18, 50, 53, 55, 60] [6, 12, 21, 24, 56, 59, 62, 67] [7, 13, 20, 22, 57, 60, 63, 68] [8, 14, 21, 23, 58, 61, 64, 69] [9, 16, 26, 29, 56, 59, 66, 90] [10, 17, 25, 27, 52, 57, 60, 67] [11, 18, 26, 28, 53, 58, 61, 68] [12, 20, 30, 33, 62, 66, 70, 76] [0, 13, 21, 31, 63, 67, 71, 77] [14, 22, 30, 32, 64, 68, 72, 78] [15, 23, 31, 33, 65, 69, 73, 79] [16, 25, 34, 37, 62, 66, 75, 95] [1, 17, 26, 35, 56, 63, 67, 76] [18, 27, 34, 36, 57, 64, 68, 77] [19, 28, 35, 37, 58, 65, 69, 78] [0, 3, 20, 40, 50, 70, 75, 84] [2, 21, 30, 38, 71, 76, 80, 85] [3, 22, 31, 39, 72, 77, 81, 86] [23, 32, 38, 40, 73, 78, 82, 87] [2, 24, 33, 39, 51, 74, 79, 83] [1, 5, 25, 43, 51, 70, 75, 98] [4, 26, 34, 41, 62, 71, 76, 84] [5, 27, 35, 42, 63, 72, 77, 85] [28, 36, 41, 43, 64, 73, 78, 86] [4, 29, 37, 42, 65, 74, 79, 87] [3, 6, 8, 30, 53, 80, 84, 91] [7, 31, 38, 44, 81, 85, 88, 92] [8, 32, 39, 45, 82, 86, 89, 93] [6, 33, 40, 44, 54, 83, 87, 90] [5, 9, 11, 34, 55, 70, 80, 84] [10, 35, 41, 46, 71, 81, 85, 91] [11, 36, 42, 47, 72, 82, 86, 92] [9, 37, 43, 46, 73, 83, 87, 93] [8, 13, 15, 38, 58, 88, 91, 96] [14, 39, 44, 48, 89, 92, 94, 97] [12, 15, 40, 45, 59, 90, 93, 95] [11, 17, 19, 41, 61, 80, 88, 91] [18, 42, 46, 49, 81, 89, 92, 96] [16, 19, 43, 47, 82, 90, 93, 97] [15, 22, 24, 44, 65, 94, 96, 99] [20, 23, 45, 48, 66, 95, 97, 98] [19, 27, 29, 46, 69, 88, 94, 96] [25, 28, 47, 49, 89, 95, 97, 99] [0, 24, 32, 48, 74, 75, 98, 99] [1, 29, 36, 49, 79, 94, 98, 99]
Code ID 100-4-14 · download JSON · raw on GitHub