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[[100,24,5]] d =
n
100
k
24
d
5
kd²/n
6.0
w
8
X/Z
1
g
0.0044
r
6.0828
layers
2
swaps
425

Share this result

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)
[66, 76, 85, 92, 97]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[23, 39, 43, 84, 99]
certificate exact, d = 5 · scipy/HiGHS MILP
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 2.2361 · Z 5.3852 (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 = 6.083
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 425 nearest-neighbor SWAPs per round in total, at most 9 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_k24.txt, line 3. nonidentity supports a=[4, 5, 11], b=[9, 10, 28].
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

[[100,24,5]] — 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 = 6.00.

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_k24.txt, line 3. This row is SmallGroup(50,3) with nonidentity GAP supports a=[4, 5, 11], b=[9, 10, 28]. 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=24), 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=[4, 5, 11], b=[9, 10, 28] 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) = 24. 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, 24, 25, 34, 50, 65, 66, 85] [1, 5, 20, 48, 51, 62, 69, 81] [1, 2, 5, 33, 52, 73, 75, 91] [0, 3, 34, 41, 53, 74, 76, 97] [0, 4, 10, 24, 54, 70, 78, 88] [5, 11, 20, 30, 55, 71, 79, 95] [4, 6, 10, 40, 51, 56, 61, 82] [2, 5, 7, 11, 57, 83, 84, 99] [3, 8, 41, 46, 50, 58, 66, 92] [2, 9, 17, 33, 50, 58, 59, 86] [0, 3, 10, 18, 60, 80, 87, 98] [11, 19, 30, 38, 51, 61, 62, 89] [9, 12, 17, 45, 54, 62, 68, 89] [6, 10, 13, 18, 55, 63, 79, 90] [7, 11, 14, 19, 52, 64, 75, 96] [8, 15, 46, 49, 53, 65, 85, 97] [6, 16, 26, 40, 52, 64, 66, 92] [2, 7, 17, 27, 53, 67, 74, 93] [3, 8, 18, 28, 54, 68, 70, 94] [19, 29, 38, 44, 55, 69, 81, 95] [16, 20, 26, 48, 59, 70, 77, 94] [12, 17, 21, 27, 60, 71, 87, 95] [13, 18, 22, 28, 51, 56, 69, 72] [14, 19, 23, 29, 57, 73, 91, 99] [15, 24, 25, 49, 58, 74, 76, 92] [12, 25, 34, 45, 56, 72, 75, 96] [6, 13, 26, 35, 57, 76, 83, 97] [7, 14, 27, 36, 50, 59, 65, 77] [8, 15, 28, 37, 60, 78, 88, 98] [1, 29, 44, 48, 61, 71, 79, 89] [20, 26, 30, 35, 67, 80, 93, 98] [21, 27, 31, 36, 54, 62, 78, 81] [22, 28, 32, 37, 61, 63, 79, 82] [1, 23, 29, 33, 64, 83, 84, 96] [12, 21, 34, 41, 63, 84, 90, 99] [13, 22, 35, 42, 52, 66, 73, 85] [14, 23, 36, 43, 58, 67, 74, 86] [4, 15, 24, 37, 68, 80, 87, 94] [30, 35, 38, 42, 59, 70, 86, 88] [31, 36, 39, 43, 68, 71, 87, 89] [4, 32, 37, 40, 55, 69, 72, 90] [21, 31, 41, 46, 56, 75, 82, 91] [22, 32, 42, 47, 64, 76, 83, 92] [9, 23, 33, 43, 53, 65, 77, 93] [38, 42, 44, 47, 77, 80, 93, 94] [9, 39, 43, 45, 60, 78, 81, 95] [31, 39, 46, 49, 72, 84, 90, 96] [16, 32, 40, 47, 57, 73, 85, 97] [16, 44, 47, 48, 67, 86, 88, 98] [25, 39, 45, 49, 63, 82, 91, 99]
H_Z (50 checks, sparse supports)
[0, 8, 9, 27, 50, 53, 54, 60] [1, 6, 11, 22, 51, 52, 79, 83] [2, 14, 16, 35, 52, 57, 59, 67] [3, 15, 17, 43, 53, 58, 60, 68] [4, 12, 18, 31, 54, 56, 87, 90] [5, 13, 19, 40, 51, 52, 55, 57] [6, 22, 25, 41, 56, 63, 66, 76] [7, 23, 26, 47, 57, 64, 67, 77] [8, 9, 24, 36, 58, 65, 68, 78] [9, 20, 27, 38, 59, 62, 93, 95] [10, 21, 28, 45, 54, 56, 60, 63] [6, 11, 29, 32, 55, 57, 61, 64] [1, 11, 12, 31, 62, 71, 75, 84] [13, 32, 34, 49, 63, 72, 76, 85] [14, 16, 33, 42, 64, 73, 77, 86] [0, 15, 27, 43, 65, 74, 78, 87] [0, 8, 16, 35, 66, 70, 97, 98] [17, 30, 36, 48, 59, 62, 67, 71] [12, 18, 37, 39, 60, 63, 68, 72] [1, 19, 22, 40, 61, 64, 69, 73] [4, 18, 20, 38, 51, 55, 70, 80] [5, 21, 29, 39, 71, 81, 84, 91] [22, 25, 40, 46, 72, 82, 85, 92] [2, 23, 35, 47, 73, 83, 86, 93] [3, 17, 24, 36, 50, 54, 74, 87] [2, 14, 25, 41, 50, 74, 75, 99] [3, 24, 26, 42, 66, 70, 76, 80] [20, 27, 43, 44, 67, 71, 77, 81] [4, 28, 31, 45, 68, 72, 78, 82] [5, 13, 29, 32, 69, 73, 79, 83] [10, 30, 37, 44, 55, 61, 80, 88] [1, 19, 31, 45, 81, 89, 91, 96] [6, 32, 41, 49, 82, 90, 92, 97] [7, 26, 33, 42, 52, 59, 83, 93] [7, 33, 34, 46, 50, 53, 75, 84] [0, 15, 35, 47, 76, 80, 85, 88] [9, 36, 38, 48, 77, 81, 86, 89] [10, 21, 37, 39, 78, 82, 87, 90] [4, 28, 38, 48, 61, 69, 88, 94] [11, 12, 29, 39, 89, 95, 96, 99] [13, 34, 40, 46, 56, 66, 90, 97] [2, 23, 41, 49, 53, 58, 84, 91] [8, 16, 24, 42, 85, 88, 92, 94] [17, 30, 43, 44, 86, 89, 93, 95] [18, 20, 37, 44, 69, 79, 94, 98] [5, 19, 21, 45, 62, 75, 95, 99] [14, 25, 33, 46, 58, 65, 91, 96] [3, 15, 26, 47, 92, 94, 97, 98] [10, 28, 30, 48, 51, 70, 79, 98] [7, 23, 34, 49, 65, 74, 96, 99]
Code ID 100-24-5 · download JSON · raw on GitHub