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[[104,8,12]] d ≤
n
104
k
8
d
12
kd²/n
11.077
w
8
X/Z
1
g
0.0055
r
6.7082
layers
2
swaps
705

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[1, 29, 30, 32, 35, 38, 66, 70, 83, 85, 86, 89]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[0, 2, 3, 6, 9, 12, 13, 16, 91, 94, 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 (2,6): 624 (2,8): 130 (2,10): 520 (3,2): 52 (3,4): 52 (3,6): 3640 (3,8): 2704 (3,10): 12532 (3,12): 4368 (3,14): 5304 (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 (2,6): 624 (2,8): 130 (2,10): 520 (3,2): 52 (3,4): 52 (3,6): 3640 (3,8): 2704 (3,10): 12532 (3,12): 4368 (3,14): 5304 (3,16): 312
witness diameter X 6.0 · Z 6.4031 (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 = 6.708
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 705 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(52,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order52_k8.txt, line 8. nonidentity supports a=[4], b=[2, 3, 10, 12, 31].
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

[[104,8,12]] — 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 = 11.08.

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_k8.txt, line 8. This row is SmallGroup(52,1) with nonidentity GAP supports a=[4], b=[2, 3, 10, 12, 31]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 12, 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=[4], b=[2, 3, 10, 12, 31] 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) = 8. 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, 47, 52, 54, 56, 64, 78, 91] [1, 5, 52, 53, 56, 59, 80, 93] [2, 50, 52, 53, 54, 61, 75, 94] [0, 3, 55, 58, 60, 82, 95, 103] [4, 8, 53, 54, 56, 62, 77, 96] [5, 9, 55, 57, 60, 84, 97, 99] [2, 6, 55, 57, 58, 79, 98, 101] [3, 7, 56, 59, 62, 86, 99, 100] [8, 12, 57, 58, 60, 81, 100, 102] [9, 13, 52, 61, 64, 88, 95, 101] [6, 10, 53, 59, 62, 83, 97, 102] [7, 11, 52, 63, 66, 90, 96, 103] [12, 16, 54, 61, 64, 85, 98, 103] [13, 17, 53, 65, 68, 91, 92, 99] [10, 14, 54, 63, 66, 87, 93, 101] [11, 15, 55, 67, 70, 92, 94, 100] [16, 20, 56, 65, 68, 89, 94, 102] [17, 21, 57, 69, 72, 87, 95, 96] [14, 18, 58, 67, 70, 89, 91, 97] [15, 19, 59, 71, 74, 88, 96, 98] [20, 24, 60, 69, 72, 90, 93, 98] [21, 25, 61, 73, 76, 83, 91, 100] [18, 22, 62, 71, 74, 85, 93, 95] [19, 23, 63, 75, 78, 84, 92, 102] [24, 28, 64, 73, 76, 86, 94, 97] [25, 29, 65, 77, 79, 80, 87, 103] [22, 26, 66, 75, 78, 81, 89, 99] [23, 27, 54, 67, 79, 80, 82, 88] [28, 32, 68, 77, 80, 82, 90, 101] [29, 33, 56, 69, 75, 81, 83, 84] [26, 30, 52, 70, 77, 79, 82, 85] [27, 31, 58, 71, 76, 83, 84, 86] [32, 36, 53, 72, 78, 81, 84, 86] [33, 37, 60, 71, 73, 79, 85, 88] [30, 34, 55, 73, 74, 81, 83, 86] [31, 35, 62, 72, 75, 80, 87, 90] [36, 40, 57, 74, 76, 82, 85, 88] [37, 41, 64, 67, 75, 77, 89, 92] [34, 38, 59, 69, 77, 78, 87, 90] [35, 39, 66, 68, 76, 79, 91, 94] [40, 44, 61, 70, 78, 80, 89, 92] [41, 45, 63, 68, 71, 81, 93, 96] [38, 42, 63, 65, 73, 82, 91, 94] [39, 43, 64, 70, 72, 83, 95, 98] [44, 48, 65, 66, 74, 84, 93, 96] [45, 49, 59, 67, 72, 85, 97, 100] [42, 46, 61, 67, 69, 86, 95, 98] [43, 47, 60, 68, 74, 87, 99, 102] [48, 51, 62, 69, 70, 88, 97, 100] [1, 49, 55, 63, 76, 89, 101, 103] [46, 50, 57, 65, 71, 90, 99, 102] [4, 51, 58, 66, 73, 92, 101, 103]
H_Z (52 checks, sparse supports)
[0, 1, 2, 9, 11, 30, 52, 55] [1, 2, 4, 10, 13, 32, 53, 101] [0, 2, 4, 12, 14, 27, 54, 58] [3, 5, 6, 15, 34, 49, 55, 59] [0, 1, 4, 7, 16, 29, 56, 103] [5, 6, 8, 17, 36, 50, 53, 57] [3, 6, 8, 18, 31, 51, 58, 62] [1, 7, 10, 19, 38, 45, 59, 63] [3, 5, 8, 20, 33, 47, 56, 60] [2, 9, 12, 21, 40, 46, 57, 61] [4, 7, 10, 22, 35, 48, 62, 66] [11, 14, 23, 41, 42, 49, 63, 67] [0, 9, 12, 24, 37, 43, 60, 64] [13, 16, 25, 42, 44, 50, 61, 65] [11, 14, 26, 39, 44, 51, 66, 70] [15, 18, 27, 37, 45, 46, 67, 71] [13, 16, 28, 39, 41, 47, 64, 68] [17, 20, 29, 38, 46, 48, 65, 69] [15, 18, 30, 40, 43, 48, 70, 74] [19, 22, 31, 33, 41, 50, 71, 75] [17, 20, 32, 35, 43, 45, 68, 72] [21, 24, 33, 34, 42, 51, 69, 73] [19, 22, 34, 36, 44, 47, 74, 78] [2, 23, 26, 29, 35, 37, 75, 79] [21, 24, 31, 36, 39, 49, 72, 76] [4, 25, 28, 30, 37, 38, 73, 77] [0, 23, 26, 32, 38, 40, 78, 82] [6, 25, 27, 30, 33, 39, 79, 83] [1, 25, 27, 28, 35, 40, 76, 80] [8, 26, 29, 32, 34, 41, 77, 81] [3, 27, 28, 30, 36, 42, 82, 86] [10, 21, 29, 31, 34, 43, 83, 87] [5, 23, 29, 31, 32, 44, 80, 84] [12, 22, 30, 33, 36, 45, 81, 85] [7, 24, 31, 32, 34, 46, 86, 90] [14, 17, 25, 35, 38, 47, 87, 91] [9, 19, 27, 33, 36, 48, 84, 88] [16, 18, 26, 37, 40, 49, 85, 89] [11, 20, 28, 35, 38, 50, 90, 94] [0, 13, 18, 21, 39, 42, 91, 95] [13, 15, 23, 37, 40, 51, 88, 92] [1, 14, 20, 22, 41, 44, 89, 93] [2, 15, 16, 24, 39, 42, 94, 98] [3, 9, 17, 22, 43, 46, 95, 99] [4, 11, 17, 19, 41, 44, 92, 96] [5, 10, 18, 24, 45, 48, 93, 97] [6, 12, 19, 20, 43, 46, 98, 102] [5, 7, 13, 26, 47, 50, 52, 99] [7, 8, 15, 21, 45, 48, 96, 100] [6, 9, 14, 28, 49, 51, 97, 101] [8, 10, 16, 23, 47, 50, 54, 102] [3, 11, 12, 25, 49, 51, 100, 103]
Code ID 104-8-12 · download JSON · raw on GitHub