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[[36,2,8]] d =
n
36
k
2
d
8
kd²/n
3.556
w
8
X/Z
1
g
0.0053
r
5.099
layers
2
swaps
95

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[0, 7, 8, 17, 20, 23, 25, 28]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[1, 5, 6, 15, 17, 19, 26, 29]
certificate exact, d = 8 · scipy/HiGHS MILP
X: no logical < 8 exists; Z: no logical < 8 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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×18 (2,2)×18 (3,2)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 18 (1,6): 18 (2,2): 18 (2,4): 36 (2,6): 162 (2,8): 81 (2,10): 54 (3,2): 18 (3,4): 324 (3,6): 1116 (3,8): 1386 (3,10): 1008 (3,12): 114 (3,14): 36
trapping sets H_Z (1,2)×18 (2,2)×18 (3,2)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 18 (1,6): 18 (2,2): 18 (2,4): 36 (2,6): 162 (2,8): 81 (2,10): 54 (3,2): 18 (3,4): 324 (3,6): 1116 (3,8): 1386 (3,10): 1008 (3,12): 114 (3,14): 36
witness diameter X 5.3852 · Z 5.099 (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 = 5.099
X checkZ checkqubit site (18)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 95 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(18,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order18_k2.txt, line 1. nonidentity supports a=[5], b=[2, 3, 8, 10, 16].
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

[[36,2,8]] — two-block group-algebra code on SmallGroup(18,2)

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 = 3.56.

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_wt8wtL2_order18_k2.txt, line 1. This row is SmallGroup(18,2) with nonidentity GAP supports a=[5], b=[2, 3, 8, 10, 16]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, 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(18,2), 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], b=[2, 3, 8, 10, 16] 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 = 36 - rank(HX) - rank(HZ) = 2. 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 18 (max weight 8) · Z-checks 18 (max weight 8)
H_X (18 checks, sparse supports)
[0, 17, 18, 19, 27, 30, 33, 34] [1, 16, 18, 19, 24, 31, 32, 35] [1, 2, 18, 20, 22, 23, 26, 35] [3, 9, 21, 22, 23, 24, 32, 34] [0, 4, 19, 20, 21, 22, 29, 34] [5, 6, 20, 21, 23, 27, 30, 35] [4, 6, 19, 20, 24, 27, 28, 31] [5, 7, 18, 21, 25, 27, 28, 29] [8, 14, 24, 26, 28, 29, 30, 35] [2, 9, 18, 22, 24, 25, 27, 33] [3, 10, 19, 23, 24, 25, 26, 28] [11, 12, 25, 26, 27, 29, 32, 34] [10, 12, 20, 23, 25, 30, 32, 33] [11, 13, 19, 21, 26, 31, 32, 33] [7, 14, 21, 22, 28, 30, 31, 32] [8, 15, 18, 23, 29, 30, 31, 33] [15, 16, 22, 25, 29, 31, 34, 35] [13, 17, 20, 26, 28, 33, 34, 35]
H_Z (18 checks, sparse supports)
[0, 1, 2, 7, 9, 15, 18, 22] [0, 1, 4, 6, 10, 13, 19, 20] [2, 4, 5, 6, 12, 17, 20, 27] [3, 4, 5, 7, 13, 14, 21, 28] [2, 3, 4, 9, 14, 16, 22, 24] [2, 3, 5, 10, 12, 15, 23, 25] [1, 3, 6, 8, 9, 10, 23, 24] [7, 9, 10, 11, 12, 16, 25, 32] [2, 8, 10, 11, 13, 17, 26, 33] [0, 5, 6, 7, 9, 11, 21, 27] [6, 7, 8, 10, 14, 17, 28, 30] [4, 7, 8, 11, 15, 16, 29, 31] [0, 5, 8, 12, 14, 15, 29, 30] [1, 6, 13, 14, 15, 16, 31, 35] [1, 3, 11, 12, 13, 14, 26, 32] [0, 9, 12, 13, 15, 17, 33, 34] [0, 3, 4, 11, 16, 17, 19, 34] [1, 2, 5, 8, 16, 17, 18, 35]
Code ID 36-2-8 · download JSON · raw on GitHub