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[[112,16,6]] d =
n
112
k
16
d
6
kd²/n
5.143
w
6
X/Z
1
g
0.0129
r
4.4721
layers
2
swaps
223

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[16, 40, 54, 70, 80, 86]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[7, 33, 49, 63, 79, 95]
certificate exact, d = 6 · scipy/HiGHS MILP
X: no logical < 6 exists; Z: no logical < 6 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 6 · H_Z 6 (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 (2,2)×56 (3,2)×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 (2,2): 56 (2,4): 448 (2,6): 336 (3,2): 112 (3,4): 1904 (3,6): 4704 (3,8): 2520 (3,10): 224
trapping sets H_Z (1,2)×56 (2,2)×56 (3,2)×112 (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 (2,2): 56 (2,4): 448 (2,6): 336 (3,2): 112 (3,4): 1904 (3,6): 4704 (3,8): 2520 (3,10): 224
witness diameter X 2.8284 · Z 3.1623 (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 = 4.472
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 223 nearest-neighbor SWAPs per round in total, at most 5 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 contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[112,16,6]] — two-block group-algebra code on SmallGroup(56,1)

Direction & hypothesis

Target: unrestricted weight-6 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 = 5.14.

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_k16.txt, line 3. This row is SmallGroup(56,1) with nonidentity GAP supports a=[5], b=[2, 12, 24]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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

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