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[[108,40,2]] d =
n
108
k
40
d
2
kd²/n
1.481
w
7
X/Z
1
g
0.0088
r
3.6056
layers
2
swaps
102

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[93, 106]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[37, 46]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×54 (2,0)×81 (3,3)×342 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 54 (1,4): 54 (2,0): 81 (2,4): 108 (2,5): 648 (3,3): 342 (3,4): 972 (3,5): 1296 (3,6): 2916 (3,7): 3240
trapping sets H_Z (1,3)×54 (2,0)×81 (3,3)×342 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 54 (1,4): 54 (2,0): 81 (2,4): 108 (2,5): 648 (3,3): 342 (3,4): 972 (3,5): 1296 (3,6): 2916 (3,7): 3240
witness diameter X 1.4142 · Z 1.0 (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 = 3.606
X checkZ checkqubit site (54)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 102 nearest-neighbor SWAPs per round in total, at most 3 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(54,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order54_k40.txt, line 1. nonidentity supports a=[3, 9], b=[2, 4, 7].
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

[[108,40,2]] — two-block group-algebra code on SmallGroup(54,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.48.

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_wt7wtL3_order54_k40.txt, line 1. This row is SmallGroup(54,3) with nonidentity GAP supports a=[3, 9], b=[2, 4, 7]. 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=40), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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(54,3), 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=[3, 9], b=[2, 4, 7] 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 = 108 - rank(HX) - rank(HZ) = 40. 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 54 (max weight 7) · Z-checks 54 (max weight 7)
H_X (54 checks, sparse supports)
[0, 2, 8, 54, 55, 60, 79] [1, 5, 14, 54, 55, 57, 86] [0, 2, 8, 56, 59, 69, 91] [3, 9, 20, 54, 55, 57, 86] [4, 10, 21, 58, 73, 87, 93] [1, 5, 14, 56, 59, 63, 97] [6, 15, 27, 54, 55, 60, 79] [7, 16, 28, 61, 67, 80, 99] [0, 2, 8, 62, 68, 81, 100] [3, 9, 20, 56, 59, 63, 97] [4, 10, 21, 64, 85, 98, 102] [11, 22, 34, 57, 65, 72, 86] [12, 23, 35, 58, 66, 71, 73] [13, 24, 36, 61, 65, 67, 72] [1, 5, 14, 62, 68, 74, 103] [6, 15, 27, 56, 59, 69, 91] [7, 16, 28, 70, 78, 92, 105] [17, 29, 40, 60, 66, 71, 79] [18, 30, 41, 61, 65, 67, 72] [19, 31, 42, 58, 66, 71, 73] [3, 9, 20, 62, 68, 74, 103] [4, 10, 21, 75, 96, 104, 106] [11, 22, 34, 63, 76, 84, 97] [12, 23, 35, 64, 77, 83, 85] [13, 24, 36, 70, 76, 78, 84] [25, 37, 46, 60, 66, 71, 79] [26, 38, 47, 61, 67, 80, 99] [6, 15, 27, 62, 68, 81, 100] [7, 16, 28, 82, 90, 101, 107] [17, 29, 40, 69, 77, 83, 91] [18, 30, 41, 70, 76, 78, 84] [19, 31, 42, 64, 77, 83, 85] [32, 43, 49, 57, 65, 72, 86] [33, 44, 50, 58, 73, 87, 93] [11, 22, 34, 74, 88, 95, 103] [12, 23, 35, 75, 89, 94, 96] [13, 24, 36, 82, 88, 90, 95] [25, 37, 46, 69, 77, 83, 91] [26, 38, 47, 70, 78, 92, 105] [39, 48, 52, 80, 87, 93, 99] [17, 29, 40, 81, 89, 94, 100] [18, 30, 41, 82, 88, 90, 95] [19, 31, 42, 75, 89, 94, 96] [32, 43, 49, 63, 76, 84, 97] [33, 44, 50, 64, 85, 98, 102] [45, 51, 53, 80, 87, 93, 99] [25, 37, 46, 81, 89, 94, 100] [26, 38, 47, 82, 90, 101, 107] [39, 48, 52, 92, 98, 102, 105] [32, 43, 49, 74, 88, 95, 103] [33, 44, 50, 75, 96, 104, 106] [45, 51, 53, 92, 98, 102, 105] [39, 48, 52, 101, 104, 106, 107] [45, 51, 53, 101, 104, 106, 107]
H_Z (54 checks, sparse supports)
[0, 1, 3, 6, 54, 56, 62] [0, 1, 3, 6, 55, 59, 68] [2, 5, 9, 15, 54, 56, 62] [1, 3, 11, 32, 57, 63, 74] [4, 12, 19, 33, 58, 64, 75] [2, 5, 9, 15, 55, 59, 68] [0, 6, 17, 25, 60, 69, 81] [7, 13, 18, 26, 61, 70, 82] [8, 14, 20, 27, 54, 56, 62] [5, 9, 22, 43, 57, 63, 74] [10, 23, 31, 44, 58, 64, 75] [11, 13, 18, 32, 65, 76, 88] [12, 17, 19, 25, 66, 77, 89] [7, 13, 18, 26, 67, 78, 90] [8, 14, 20, 27, 55, 59, 68] [2, 15, 29, 37, 60, 69, 81] [16, 24, 30, 38, 61, 70, 82] [12, 17, 19, 25, 71, 83, 94] [11, 13, 18, 32, 72, 84, 95] [4, 12, 19, 33, 73, 85, 96] [14, 20, 34, 49, 57, 63, 74] [21, 35, 42, 50, 58, 64, 75] [22, 24, 30, 43, 65, 76, 88] [23, 29, 31, 37, 66, 77, 89] [16, 24, 30, 38, 67, 78, 90] [0, 6, 17, 25, 79, 91, 100] [7, 26, 39, 45, 80, 92, 101] [8, 27, 40, 46, 60, 69, 81] [28, 36, 41, 47, 61, 70, 82] [23, 29, 31, 37, 71, 83, 94] [22, 24, 30, 43, 72, 84, 95] [10, 23, 31, 44, 73, 85, 96] [1, 3, 11, 32, 86, 97, 103] [4, 33, 39, 45, 87, 98, 104] [34, 36, 41, 49, 65, 76, 88] [35, 40, 42, 46, 66, 77, 89] [28, 36, 41, 47, 67, 78, 90] [2, 15, 29, 37, 79, 91, 100] [16, 38, 48, 51, 80, 92, 101] [4, 33, 39, 45, 93, 102, 106] [35, 40, 42, 46, 71, 83, 94] [34, 36, 41, 49, 72, 84, 95] [21, 35, 42, 50, 73, 85, 96] [5, 9, 22, 43, 86, 97, 103] [10, 44, 48, 51, 87, 98, 104] [7, 26, 39, 45, 99, 105, 107] [8, 27, 40, 46, 79, 91, 100] [28, 47, 52, 53, 80, 92, 101] [10, 44, 48, 51, 93, 102, 106] [14, 20, 34, 49, 86, 97, 103] [21, 50, 52, 53, 87, 98, 104] [16, 38, 48, 51, 99, 105, 107] [21, 50, 52, 53, 93, 102, 106] [28, 47, 52, 53, 99, 105, 107]
Code ID 108-40-2 · download JSON · raw on GitHub