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[[100,8,5]] d =
n
100
k
8
d
5
kd²/n
2.0
w
5
X/Z
1
g
0.005
r
4.4721
layers
2
swaps
343

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 5, w_Z = 5 (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)
[56, 65, 71, 83, 88]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[6, 15, 21, 33, 38]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×50 (2,2)×50 (3,2)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 50 (1,3): 50 (2,2): 50 (2,3): 300 (2,4): 150 (3,2): 50 (3,3): 900 (3,4): 1650 (3,5): 750 (3,6): 300 (3,7): 50
trapping sets H_Z (1,2)×50 (2,2)×50 (3,2)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 50 (1,3): 50 (2,2): 50 (2,3): 300 (2,4): 150 (3,2): 50 (3,3): 900 (3,4): 1650 (3,5): 750 (3,6): 300 (3,7): 50
witness diameter X 6.7082 · Z 7.6158 (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 (52)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 343 nearest-neighbor SWAPs per round in total, at most 6 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_wt5wtL2_order50_k8.txt, line 2. nonidentity supports a=[8], b=[5, 14].
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

[[100,8,5]] — two-block group-algebra code on SmallGroup(50,3)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.00.

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_wt5wtL2_order50_k8.txt, line 2. This row is SmallGroup(50,3) with nonidentity GAP supports a=[8], b=[5, 14]. 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=8), 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 (d=5) 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=[8], b=[5, 14] 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) = 8. 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 5) · Z-checks 50 (max weight 5)
H_X (50 checks, sparse supports)
[0, 48, 50, 75, 95] [1, 34, 51, 70, 97] [2, 24, 51, 52, 98] [3, 20, 53, 62, 99] [4, 5, 50, 54, 99] [5, 41, 55, 66, 98] [6, 33, 54, 56, 74] [0, 7, 57, 70, 79] [8, 30, 58, 71, 96] [9, 10, 52, 59, 79] [10, 11, 60, 74, 75] [11, 46, 61, 76, 94] [12, 40, 59, 62, 83] [2, 13, 50, 63, 87] [3, 14, 64, 69, 80] [15, 38, 65, 81, 91] [16, 17, 56, 66, 87] [17, 18, 51, 67, 83] [18, 19, 65, 68, 84] [19, 49, 69, 85, 88] [20, 45, 66, 70, 90] [6, 21, 52, 71, 93] [7, 22, 53, 72, 78] [8, 23, 61, 73, 88] [24, 44, 74, 84, 89] [25, 26, 62, 75, 93] [26, 27, 54, 76, 90] [27, 28, 55, 73, 77] [28, 29, 58, 78, 91] [25, 29, 79, 80, 92] [12, 30, 56, 80, 97] [13, 31, 57, 81, 86] [14, 32, 58, 68, 82] [15, 33, 55, 83, 94] [34, 35, 59, 84, 95] [35, 36, 60, 82, 85] [36, 37, 61, 64, 86] [1, 37, 53, 87, 96] [21, 38, 63, 88, 92] [22, 39, 64, 77, 89] [23, 40, 60, 65, 90] [41, 42, 67, 89, 91] [42, 43, 68, 72, 92] [4, 43, 57, 69, 93] [31, 44, 72, 85, 94] [32, 45, 67, 73, 95] [46, 47, 77, 81, 96] [9, 47, 63, 78, 97] [39, 48, 76, 82, 98] [16, 49, 71, 86, 99]
H_Z (50 checks, sparse supports)
[0, 4, 13, 50, 57] [1, 2, 17, 51, 87] [2, 9, 21, 52, 63] [3, 22, 37, 53, 64] [4, 6, 26, 54, 93] [5, 27, 33, 54, 55] [6, 16, 30, 56, 71] [7, 31, 43, 57, 72] [8, 28, 32, 58, 73] [9, 12, 34, 59, 97] [10, 35, 40, 59, 60] [11, 23, 36, 60, 61] [3, 12, 25, 62, 80] [13, 38, 47, 63, 81] [14, 36, 39, 64, 82] [15, 18, 40, 65, 83] [5, 16, 20, 66, 99] [17, 41, 45, 66, 67] [18, 32, 42, 67, 68] [14, 19, 43, 68, 69] [1, 7, 20, 53, 70] [8, 21, 49, 71, 88] [22, 42, 44, 72, 89] [23, 27, 45, 73, 90] [6, 10, 24, 52, 74] [0, 10, 25, 75, 79] [11, 26, 48, 75, 76] [27, 39, 46, 76, 77] [22, 28, 47, 77, 78] [7, 9, 29, 78, 79] [14, 29, 30, 58, 80] [15, 31, 46, 81, 94] [32, 35, 48, 82, 95] [12, 17, 33, 56, 83] [18, 24, 34, 51, 84] [19, 35, 44, 84, 85] [31, 36, 49, 85, 86] [13, 16, 37, 86, 87] [19, 23, 38, 65, 88] [24, 39, 41, 89, 98] [20, 26, 40, 62, 90] [15, 28, 41, 55, 91] [29, 38, 42, 91, 92] [21, 25, 43, 92, 93] [11, 33, 44, 74, 94] [0, 34, 45, 70, 95] [8, 37, 46, 61, 96] [1, 30, 47, 96, 97] [2, 5, 48, 50, 98] [3, 4, 49, 69, 99]
Code ID 100-8-5 · download JSON · raw on GitHub