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[[100,28,5]] d =
n
100
k
28
d
5
kd²/n
7.0
w
8
X/Z
1
g
0.0068
r
5.6569
layers
2
swaps
373

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[65, 73, 82, 89, 94]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[29, 37, 43, 47, 49]
certificate exact, d = 5 · scipy/HiGHS cutoff IP
X: no logical < 5 exists; Z: no logical < 5 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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×100 (2,4)×200 (3,4)×600 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 100 (2,4): 200 (2,6): 1000 (3,4): 600 (3,6): 4000 (3,8): 15000 (3,10): 800
trapping sets H_Z (1,4)×100 (2,4)×200 (3,4)×600 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 100 (2,4): 200 (2,6): 1000 (3,4): 600 (3,6): 4000 (3,8): 15000 (3,10): 800
witness diameter X 8.0623 · Z 8.0 (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 = 5.657
X checkZ checkqubit site (51)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 373 nearest-neighbor SWAPs per round in total, at most 8 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(50,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order50_k28.txt, line 4. nonidentity supports a=[3, 4, 8], b=[5, 9, 19].
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

[[100,28,5]] — two-block group-algebra code on SmallGroup(50,3)

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 = 7.00.

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_wt8wtL4_order50_k28.txt, line 4. This row is SmallGroup(50,3) with nonidentity GAP supports a=[3, 4, 8], b=[5, 9, 19]. 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=28), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 5, 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(50,3), 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, 4, 8], b=[5, 9, 19] 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 = 100 - rank(HX) - rank(HZ) = 28. 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 50 (max weight 8) · Z-checks 50 (max weight 8)
H_X (50 checks, sparse supports)
[0, 20, 24, 48, 50, 65, 75, 91] [1, 5, 25, 34, 51, 69, 70, 88] [0, 2, 24, 33, 51, 52, 61, 73] [0, 3, 20, 30, 53, 74, 84, 99] [1, 4, 5, 10, 50, 54, 58, 78] [5, 11, 34, 41, 55, 79, 80, 98] [2, 6, 33, 40, 54, 56, 68, 82] [0, 2, 3, 7, 55, 57, 79, 83] [3, 8, 30, 38, 50, 58, 75, 96] [4, 9, 10, 17, 52, 59, 64, 86] [5, 10, 11, 18, 53, 60, 74, 87] [11, 19, 41, 46, 51, 61, 70, 94] [6, 12, 40, 45, 59, 62, 77, 89] [2, 6, 7, 13, 60, 63, 87, 90] [3, 7, 8, 14, 51, 52, 64, 69] [8, 15, 38, 44, 53, 65, 91, 99] [9, 16, 17, 26, 56, 66, 72, 92] [10, 17, 18, 27, 57, 67, 83, 93] [11, 18, 19, 28, 50, 54, 65, 68] [19, 29, 46, 49, 55, 69, 88, 98] [12, 20, 45, 48, 66, 70, 85, 94] [6, 12, 13, 21, 67, 71, 93, 95] [7, 13, 14, 22, 54, 56, 72, 78] [8, 14, 15, 23, 57, 61, 73, 79] [15, 24, 44, 48, 58, 74, 84, 96] [16, 25, 26, 34, 62, 75, 81, 96] [17, 26, 27, 35, 63, 76, 90, 97] [18, 27, 28, 36, 52, 59, 73, 77] [19, 28, 29, 37, 58, 60, 74, 78] [1, 25, 29, 49, 61, 79, 80, 94] [12, 20, 21, 30, 76, 80, 97, 98] [13, 21, 22, 31, 59, 62, 81, 86] [14, 22, 23, 32, 63, 68, 82, 87] [15, 23, 24, 33, 55, 64, 69, 83] [26, 34, 35, 41, 71, 84, 95, 99] [27, 35, 36, 42, 56, 66, 82, 85] [28, 36, 37, 43, 64, 67, 83, 86] [1, 4, 29, 37, 53, 65, 68, 87] [21, 30, 31, 38, 66, 70, 88, 92] [22, 31, 32, 39, 71, 77, 89, 93] [23, 32, 33, 40, 60, 72, 78, 90] [35, 41, 42, 46, 62, 75, 89, 91] [36, 42, 43, 47, 72, 76, 90, 92] [4, 9, 37, 43, 57, 73, 77, 93] [31, 38, 39, 44, 80, 85, 94, 97] [32, 39, 40, 45, 67, 81, 86, 95] [42, 46, 47, 49, 81, 84, 95, 96] [9, 16, 43, 47, 63, 82, 85, 97] [39, 44, 45, 48, 76, 88, 92, 98] [16, 25, 47, 49, 71, 89, 91, 99]
H_Z (50 checks, sparse supports)
[0, 4, 8, 18, 50, 52, 53, 57] [1, 2, 11, 14, 51, 54, 79, 87] [2, 9, 14, 27, 52, 56, 57, 63] [3, 10, 15, 37, 53, 57, 58, 64] [4, 6, 18, 22, 54, 59, 87, 93] [5, 7, 19, 33, 51, 54, 55, 60] [6, 16, 22, 35, 56, 62, 63, 71] [7, 17, 23, 43, 57, 63, 64, 72] [4, 8, 24, 28, 58, 64, 65, 73] [9, 12, 27, 31, 59, 66, 93, 97] [10, 13, 28, 40, 54, 59, 60, 67] [2, 11, 23, 29, 55, 60, 61, 68] [12, 25, 31, 41, 62, 70, 71, 80] [13, 26, 32, 47, 63, 71, 72, 81] [9, 14, 33, 36, 64, 72, 73, 82] [0, 15, 18, 37, 65, 73, 74, 83] [16, 20, 35, 38, 66, 75, 97, 99] [17, 21, 36, 45, 59, 66, 67, 76] [6, 18, 32, 37, 60, 67, 68, 77] [1, 14, 19, 33, 61, 68, 69, 78] [1, 11, 20, 38, 50, 53, 70, 80] [21, 34, 39, 49, 71, 80, 81, 88] [16, 22, 40, 42, 72, 81, 82, 89] [2, 23, 27, 43, 73, 82, 83, 90] [3, 10, 24, 28, 50, 52, 74, 83] [0, 8, 25, 41, 51, 75, 79, 99] [26, 30, 42, 48, 66, 75, 76, 84] [12, 27, 39, 43, 67, 76, 77, 85] [4, 22, 28, 40, 68, 77, 78, 86] [5, 7, 23, 29, 69, 78, 79, 87] [5, 29, 30, 44, 53, 58, 80, 88] [25, 31, 45, 46, 81, 88, 89, 94] [6, 32, 35, 47, 82, 89, 90, 95] [7, 17, 33, 36, 52, 56, 83, 90] [3, 24, 34, 46, 51, 55, 75, 84] [20, 35, 44, 47, 76, 84, 85, 91] [9, 31, 36, 45, 77, 85, 86, 92] [10, 13, 32, 37, 78, 86, 87, 93] [1, 19, 38, 48, 58, 65, 88, 94] [12, 39, 41, 49, 89, 94, 95, 98] [13, 26, 40, 42, 56, 62, 90, 95] [0, 15, 41, 49, 55, 61, 84, 91] [16, 38, 42, 48, 85, 91, 92, 96] [17, 21, 39, 43, 86, 92, 93, 97] [11, 20, 29, 44, 65, 74, 94, 98] [21, 34, 45, 46, 62, 70, 95, 98] [8, 24, 25, 46, 61, 69, 91, 96] [26, 30, 44, 47, 92, 96, 97, 99] [5, 19, 30, 48, 50, 70, 74, 98] [3, 15, 34, 49, 69, 79, 96, 99]
Code ID 100-28-5 · download JSON · raw on GitHub