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[[112,42,2]] d =
n
112
k
42
d
2
kd²/n
1.5
w
6
X/Z
1
g
0.0089
r
3.6056
layers
2
swaps
133

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 6, w_Z = 6 (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)
[106, 109]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[17, 21]
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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×56 (2,0)×56 (3,0)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 56 (1,4): 56 (2,0): 56 (2,2): 112 (2,4): 224 (2,6): 224 (3,0): 112 (3,2): 168 (3,4): 1512 (3,6): 2016 (3,8): 1344
trapping sets H_Z (1,2)×56 (2,0)×28 (3,2)×896 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 56 (1,4): 56 (2,0): 28 (2,2): 112 (2,4): 392 (3,2): 896 (3,4): 1344 (3,6): 840 (3,8): 112
witness diameter X 0.0 · Z 2.2361 (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 (56)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 133 nearest-neighbor SWAPs per round in total, at most 4 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(56,9), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order56_k42.txt, line 1. nonidentity supports a=[2], b=[2, 9, 13].
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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[112,42,2]] — two-block group-algebra code on SmallGroup(56,9)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.50.

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_wt6wtL2_order56_k42.txt, line 1. This row is SmallGroup(56,9) with nonidentity GAP supports a=[2], b=[2, 9, 13]. 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=42), 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(56,9), 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], b=[2, 9, 13] 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 = 112 - rank(HX) - rank(HZ) = 42. 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 56 (max weight 6) · Z-checks 56 (max weight 6)
H_X (56 checks, sparse supports)
[0, 1, 56, 57, 104, 111] [0, 1, 56, 57, 107, 110] [2, 5, 58, 69, 98, 102] [3, 6, 58, 59, 62, 69] [4, 7, 60, 63, 107, 110] [2, 5, 61, 65, 98, 102] [3, 6, 59, 61, 62, 65] [4, 7, 60, 63, 104, 111] [8, 12, 56, 57, 64, 77] [9, 13, 61, 65, 106, 109] [10, 14, 64, 66, 70, 77] [11, 15, 61, 65, 67, 71] [8, 12, 56, 57, 68, 73] [9, 13, 58, 69, 106, 109] [10, 14, 66, 68, 70, 73] [11, 15, 58, 67, 69, 71] [16, 20, 59, 62, 72, 85] [17, 21, 60, 63, 68, 73] [18, 22, 72, 74, 78, 85] [19, 23, 68, 73, 75, 79] [16, 20, 59, 62, 76, 81] [17, 21, 60, 63, 64, 77] [18, 22, 74, 76, 78, 81] [19, 23, 64, 75, 77, 79] [24, 28, 66, 70, 80, 93] [25, 29, 67, 71, 76, 81] [26, 30, 80, 82, 86, 93] [27, 31, 76, 81, 83, 87] [24, 28, 66, 70, 84, 89] [25, 29, 67, 71, 72, 85] [26, 30, 82, 84, 86, 89] [27, 31, 72, 83, 85, 87] [32, 36, 74, 78, 88, 101] [33, 37, 75, 79, 84, 89] [34, 38, 88, 90, 94, 101] [35, 39, 84, 89, 91, 95] [32, 36, 74, 78, 92, 97] [33, 37, 75, 79, 80, 93] [34, 38, 90, 92, 94, 97] [35, 39, 80, 91, 93, 95] [40, 44, 82, 86, 96, 108] [41, 45, 83, 87, 92, 97] [42, 46, 96, 98, 102, 108] [43, 47, 92, 97, 99, 103] [40, 44, 82, 86, 100, 105] [41, 45, 83, 87, 88, 101] [42, 46, 98, 100, 102, 105] [43, 47, 88, 99, 101, 103] [48, 51, 90, 94, 104, 111] [49, 52, 91, 95, 100, 105] [50, 53, 100, 105, 106, 109] [48, 51, 90, 94, 107, 110] [49, 52, 91, 95, 96, 108] [50, 53, 96, 106, 108, 109] [54, 55, 99, 103, 107, 110] [54, 55, 99, 103, 104, 111]
H_Z (56 checks, sparse supports)
[0, 1, 8, 12, 56, 57] [0, 1, 8, 12, 56, 57] [2, 3, 13, 15, 58, 61] [3, 6, 16, 20, 59, 62] [4, 7, 17, 21, 60, 63] [5, 6, 9, 11, 58, 61] [3, 6, 16, 20, 59, 62] [4, 7, 17, 21, 60, 63] [8, 10, 21, 23, 64, 68] [5, 6, 9, 11, 65, 69] [10, 14, 24, 28, 66, 70] [11, 15, 25, 29, 67, 71] [12, 14, 17, 19, 64, 68] [2, 3, 13, 15, 65, 69] [10, 14, 24, 28, 66, 70] [11, 15, 25, 29, 67, 71] [16, 18, 29, 31, 72, 76] [12, 14, 17, 19, 73, 77] [18, 22, 32, 36, 74, 78] [19, 23, 33, 37, 75, 79] [20, 22, 25, 27, 72, 76] [8, 10, 21, 23, 73, 77] [18, 22, 32, 36, 74, 78] [19, 23, 33, 37, 75, 79] [24, 26, 37, 39, 80, 84] [20, 22, 25, 27, 81, 85] [26, 30, 40, 44, 82, 86] [27, 31, 41, 45, 83, 87] [28, 30, 33, 35, 80, 84] [16, 18, 29, 31, 81, 85] [26, 30, 40, 44, 82, 86] [27, 31, 41, 45, 83, 87] [32, 34, 45, 47, 88, 92] [28, 30, 33, 35, 89, 93] [34, 38, 48, 51, 90, 94] [35, 39, 49, 52, 91, 95] [36, 38, 41, 43, 88, 92] [24, 26, 37, 39, 89, 93] [34, 38, 48, 51, 90, 94] [35, 39, 49, 52, 91, 95] [40, 42, 52, 53, 96, 100] [36, 38, 41, 43, 97, 101] [2, 5, 42, 46, 98, 102] [43, 47, 54, 55, 99, 103] [44, 46, 49, 50, 96, 100] [32, 34, 45, 47, 97, 101] [2, 5, 42, 46, 98, 102] [43, 47, 54, 55, 99, 103] [0, 7, 48, 55, 104, 107] [44, 46, 49, 50, 105, 108] [9, 13, 50, 53, 106, 109] [1, 4, 51, 54, 104, 107] [40, 42, 52, 53, 105, 108] [9, 13, 50, 53, 106, 109] [1, 4, 51, 54, 110, 111] [0, 7, 48, 55, 110, 111]
Code ID 112-42-2 · download JSON · raw on GitHub