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[[60,24,2]] d =
n
60
k
24
d
2
kd²/n
1.6
w
7
X/Z
1
g
0.0063
r
4.0
layers
2
swaps
94

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)
[31, 39]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[42, 45]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×30 (2,0)×45 (3,3)×190 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 30 (1,4): 30 (2,0): 45 (2,4): 60 (2,5): 360 (3,3): 190 (3,4): 540 (3,5): 720 (3,6): 1620 (3,7): 1800
trapping sets H_Z (1,3)×30 (2,0)×45 (3,3)×190 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 30 (1,4): 30 (2,0): 45 (2,4): 60 (2,5): 360 (3,3): 190 (3,4): 540 (3,5): 720 (3,6): 1620 (3,7): 1800
witness diameter X 2.2361 · Z 0.0 (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
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-02
r = 4
X checkZ checkqubit site (30)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 94 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 2BGA (arXiv:2306.16400) on SmallGroup(30,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order30_k24.txt, line 1. nonidentity supports a=[3, 7], b=[2, 4, 6].
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

[[60,24,2]] — two-block group-algebra code on SmallGroup(30,4)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.60.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order30_k24.txt, line 1. This row is SmallGroup(30,4) with nonidentity GAP supports a=[3, 7], b=[2, 4, 6]. 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 ≤ 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(30,4), 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, 7], b=[2, 4, 6] 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 = 60 - 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 30 (max weight 7) · Z-checks 30 (max weight 7)
H_X (30 checks, sparse supports)
[0, 2, 6, 30, 31, 50, 53] [1, 4, 9, 30, 31, 50, 53] [0, 2, 6, 32, 34, 55, 57] [3, 7, 12, 30, 31, 33, 35] [1, 4, 9, 32, 34, 55, 57] [5, 10, 15, 30, 31, 33, 35] [0, 2, 6, 36, 39, 58, 59] [3, 7, 12, 32, 34, 37, 40] [8, 13, 18, 33, 35, 38, 41] [1, 4, 9, 36, 39, 58, 59] [5, 10, 15, 32, 34, 37, 40] [11, 16, 21, 33, 35, 38, 41] [3, 7, 12, 36, 39, 42, 45] [8, 13, 18, 37, 40, 43, 46] [14, 19, 24, 38, 41, 44, 47] [5, 10, 15, 36, 39, 42, 45] [11, 16, 21, 37, 40, 43, 46] [17, 22, 26, 38, 41, 44, 47] [8, 13, 18, 42, 45, 48, 51] [14, 19, 24, 43, 46, 49, 52] [20, 25, 28, 44, 47, 50, 53] [11, 16, 21, 42, 45, 48, 51] [17, 22, 26, 43, 46, 49, 52] [23, 27, 29, 44, 47, 50, 53] [14, 19, 24, 48, 51, 54, 56] [20, 25, 28, 49, 52, 55, 57] [17, 22, 26, 48, 51, 54, 56] [23, 27, 29, 49, 52, 55, 57] [20, 25, 28, 54, 56, 58, 59] [23, 27, 29, 54, 56, 58, 59]
H_Z (30 checks, sparse supports)
[0, 1, 3, 5, 30, 32, 36] [0, 1, 3, 5, 31, 34, 39] [2, 4, 7, 10, 30, 32, 36] [3, 5, 8, 11, 33, 37, 42] [2, 4, 7, 10, 31, 34, 39] [3, 5, 8, 11, 35, 40, 45] [6, 9, 12, 15, 30, 32, 36] [7, 10, 13, 16, 33, 37, 42] [8, 11, 14, 17, 38, 43, 48] [6, 9, 12, 15, 31, 34, 39] [7, 10, 13, 16, 35, 40, 45] [8, 11, 14, 17, 41, 46, 51] [12, 15, 18, 21, 33, 37, 42] [13, 16, 19, 22, 38, 43, 48] [14, 17, 20, 23, 44, 49, 54] [12, 15, 18, 21, 35, 40, 45] [13, 16, 19, 22, 41, 46, 51] [14, 17, 20, 23, 47, 52, 56] [18, 21, 24, 26, 38, 43, 48] [19, 22, 25, 27, 44, 49, 54] [0, 1, 20, 23, 50, 55, 58] [18, 21, 24, 26, 41, 46, 51] [19, 22, 25, 27, 47, 52, 56] [0, 1, 20, 23, 53, 57, 59] [24, 26, 28, 29, 44, 49, 54] [2, 4, 25, 27, 50, 55, 58] [24, 26, 28, 29, 47, 52, 56] [2, 4, 25, 27, 53, 57, 59] [6, 9, 28, 29, 50, 55, 58] [6, 9, 28, 29, 53, 57, 59]
Code ID 60-24-2 · download JSON · raw on GitHub