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[[104,4,14]] d ≤
n
104
k
4
d
14
kd²/n
7.538
w
8
X/Z
1
g
0.0047
r
6.3246
layers
2
swaps
530

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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)
[15, 18, 20, 21, 42, 44, 45, 46, 65, 71, 72, 90, 94, 98]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 4, 7, 10, 11, 14, 31, 34, 35, 38, 73, 76, 97, 100]
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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×52 (2,2)×52 (3,2)×52 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 52 (1,6): 52 (2,2): 52 (2,4): 52 (2,6): 520 (2,8): 130 (2,10): 520 (3,2): 52 (3,4): 416 (3,6): 2912 (3,8): 3224 (3,10): 10192 (3,12): 3432 (3,14): 5564 (3,16): 312
trapping sets H_Z (1,2)×52 (2,2)×52 (3,2)×52 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 52 (1,6): 52 (2,2): 52 (2,4): 52 (2,6): 520 (2,8): 130 (2,10): 520 (3,2): 52 (3,4): 416 (3,6): 2912 (3,8): 3224 (3,10): 10192 (3,12): 3432 (3,14): 5564 (3,16): 312
witness diameter X 5.3852 · 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
r = 6.325
X checkZ checkqubit site (53)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 530 nearest-neighbor SWAPs per round in total, at most 10 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(52,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order52_k4.txt, line 10. nonidentity supports a=[7], b=[2, 3, 4, 13, 35].
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

[[104,4,14]] — two-block group-algebra code on SmallGroup(52,1)

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 = 7.54.

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_wt8wtL2_order52_k4.txt, line 10. This row is SmallGroup(52,1) with nonidentity GAP supports a=[7], b=[2, 3, 4, 13, 35]. 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(52,1), 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=[7], b=[2, 3, 4, 13, 35] 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 = 104 - 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 52 (max weight 8) · Z-checks 52 (max weight 8)
H_X (52 checks, sparse supports)
[0, 50, 52, 54, 56, 61, 74, 99] [1, 8, 52, 53, 56, 62, 76, 101] [2, 47, 52, 53, 54, 64, 71, 102] [2, 3, 52, 55, 57, 58, 78, 103] [4, 5, 53, 54, 56, 59, 73, 103] [5, 12, 53, 57, 58, 60, 80, 99] [0, 6, 54, 55, 58, 60, 75, 101] [6, 7, 53, 55, 59, 62, 82, 100] [8, 9, 55, 56, 57, 60, 77, 102] [9, 16, 54, 57, 61, 64, 84, 95] [3, 10, 56, 58, 59, 62, 79, 97] [10, 11, 59, 63, 66, 86, 96, 101] [12, 13, 52, 60, 61, 64, 81, 98] [13, 20, 61, 65, 68, 88, 91, 102] [7, 14, 62, 63, 66, 83, 93, 103] [14, 15, 63, 67, 70, 90, 92, 97] [16, 17, 64, 65, 68, 85, 94, 99] [17, 24, 65, 69, 72, 87, 92, 98] [11, 18, 66, 67, 70, 87, 89, 100] [18, 19, 67, 71, 74, 88, 93, 94] [20, 21, 68, 69, 72, 89, 90, 95] [21, 28, 69, 73, 76, 83, 94, 96] [15, 22, 70, 71, 74, 85, 91, 96] [22, 23, 71, 75, 78, 84, 89, 98] [24, 25, 72, 73, 76, 86, 91, 93] [25, 32, 73, 77, 79, 80, 90, 100] [19, 26, 74, 75, 78, 81, 92, 95] [26, 27, 75, 79, 80, 82, 85, 102] [28, 29, 76, 77, 80, 82, 87, 97] [29, 36, 75, 77, 81, 84, 86, 103] [23, 30, 77, 78, 79, 82, 88, 99] [30, 31, 54, 76, 79, 81, 83, 86] [32, 33, 78, 80, 81, 83, 84, 101] [33, 40, 56, 71, 81, 82, 85, 88] [27, 34, 52, 73, 82, 83, 84, 86] [34, 35, 58, 72, 77, 83, 87, 90] [36, 37, 53, 74, 79, 84, 85, 88] [37, 44, 60, 67, 78, 85, 89, 92] [31, 38, 55, 69, 80, 86, 87, 90] [38, 39, 62, 68, 73, 87, 91, 94] [40, 41, 57, 70, 75, 88, 89, 92] [41, 48, 63, 64, 74, 89, 93, 96] [35, 42, 59, 65, 76, 90, 91, 94] [42, 43, 64, 66, 69, 91, 95, 98] [44, 45, 61, 66, 71, 92, 93, 96] [45, 51, 59, 68, 70, 93, 97, 100] [39, 46, 61, 63, 72, 94, 95, 98] [46, 47, 60, 65, 70, 95, 99, 102] [48, 49, 62, 65, 67, 96, 97, 100] [4, 49, 55, 66, 72, 97, 101, 103] [43, 50, 57, 67, 68, 98, 99, 102] [1, 51, 58, 63, 69, 100, 101, 103]
H_Z (52 checks, sparse supports)
[0, 1, 2, 3, 12, 34, 52, 58] [1, 2, 4, 5, 7, 36, 53, 103] [0, 2, 4, 6, 9, 31, 54, 55] [3, 6, 7, 8, 38, 49, 55, 62] [0, 1, 4, 8, 10, 33, 56, 101] [3, 5, 8, 9, 40, 50, 56, 57] [3, 5, 6, 10, 35, 51, 58, 59] [4, 7, 10, 11, 42, 45, 59, 66] [5, 6, 8, 12, 37, 47, 53, 60] [0, 9, 12, 13, 44, 46, 60, 61] [1, 7, 10, 14, 39, 48, 62, 63] [11, 14, 15, 41, 46, 51, 63, 70] [2, 9, 12, 16, 41, 43, 57, 64] [13, 16, 17, 42, 47, 48, 64, 65] [11, 14, 18, 43, 44, 49, 66, 67] [15, 18, 19, 37, 48, 50, 67, 74] [13, 16, 20, 39, 45, 50, 61, 68] [17, 20, 21, 38, 43, 51, 68, 69] [15, 18, 22, 40, 45, 47, 70, 71] [2, 19, 22, 23, 33, 44, 71, 78] [17, 20, 24, 35, 46, 49, 65, 72] [4, 21, 24, 25, 34, 39, 72, 73] [0, 19, 22, 26, 36, 41, 74, 75] [6, 23, 26, 27, 29, 40, 75, 82] [1, 21, 24, 28, 31, 42, 69, 76] [8, 25, 28, 29, 30, 35, 76, 77] [3, 23, 26, 30, 32, 37, 78, 79] [10, 25, 27, 30, 31, 36, 79, 86] [5, 25, 27, 28, 32, 38, 73, 80] [12, 26, 29, 31, 32, 33, 80, 81] [7, 27, 28, 30, 33, 34, 82, 83] [14, 21, 31, 32, 34, 35, 83, 90] [9, 23, 29, 32, 34, 36, 77, 84] [16, 22, 27, 33, 36, 37, 84, 85] [11, 24, 29, 31, 34, 38, 86, 87] [17, 18, 28, 35, 38, 39, 87, 94] [13, 19, 30, 33, 36, 40, 81, 88] [18, 20, 23, 37, 40, 41, 88, 89] [15, 20, 25, 35, 38, 42, 90, 91] [13, 22, 24, 39, 42, 43, 91, 98] [15, 17, 26, 37, 40, 44, 85, 92] [14, 19, 24, 41, 44, 45, 92, 93] [16, 19, 21, 39, 42, 46, 94, 95] [9, 20, 26, 43, 46, 47, 95, 102] [11, 21, 22, 41, 44, 48, 89, 96] [10, 15, 28, 45, 48, 49, 96, 97] [12, 17, 23, 43, 46, 50, 98, 99] [0, 5, 16, 30, 47, 50, 54, 99] [7, 18, 25, 45, 48, 51, 93, 100] [1, 6, 11, 32, 49, 51, 100, 101] [2, 8, 13, 27, 47, 50, 52, 102] [3, 4, 14, 29, 49, 51, 97, 103]
Code ID 104-4-14 · download JSON · raw on GitHub