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[[36,16,2]] d =
n
36
k
16
d
2
kd²/n
1.778
w
7
X/Z
1
g
0.0069
r
4.0
layers
2
swaps
73

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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)
[28, 32]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[34, 35]
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)×18 (2,0)×27 (3,0)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 18 (1,4): 18 (2,0): 27 (2,4): 36 (2,5): 216 (3,0): 24 (3,3): 114 (3,4): 252 (3,5): 432 (3,6): 972 (3,7): 1080
trapping sets H_Z (1,3)×18 (2,0)×27 (3,0)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 18 (1,4): 18 (2,0): 27 (2,4): 36 (2,5): 216 (3,0): 24 (3,3): 114 (3,4): 252 (3,5): 432 (3,6): 972 (3,7): 1080
witness diameter X 1.4142 · 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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 4
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 73 nearest-neighbor SWAPs per round in total, at most 4 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(18,5), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order18_k16.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

[[36,16,2]] — two-block group-algebra code on SmallGroup(18,5)

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 = 1.78.

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_order18_k16.txt, line 1. This row is SmallGroup(18,5) 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=16), 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(18,5), 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 = 36 - rank(HX) - rank(HZ) = 16. 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 7) · Z-checks 18 (max weight 7)
H_X (18 checks, sparse supports)
[0, 2, 6, 18, 19, 26, 29] [1, 4, 9, 18, 19, 26, 29] [0, 2, 6, 20, 22, 31, 33] [3, 7, 12, 18, 19, 21, 23] [1, 4, 9, 20, 22, 31, 33] [5, 10, 14, 18, 19, 21, 23] [0, 2, 6, 24, 27, 34, 35] [3, 7, 12, 20, 22, 25, 28] [8, 13, 16, 21, 23, 26, 29] [1, 4, 9, 24, 27, 34, 35] [5, 10, 14, 20, 22, 25, 28] [11, 15, 17, 21, 23, 26, 29] [3, 7, 12, 24, 27, 30, 32] [8, 13, 16, 25, 28, 31, 33] [5, 10, 14, 24, 27, 30, 32] [11, 15, 17, 25, 28, 31, 33] [8, 13, 16, 30, 32, 34, 35] [11, 15, 17, 30, 32, 34, 35]
H_Z (18 checks, sparse supports)
[0, 1, 3, 5, 18, 20, 24] [0, 1, 3, 5, 19, 22, 27] [2, 4, 7, 10, 18, 20, 24] [3, 5, 8, 11, 21, 25, 30] [2, 4, 7, 10, 19, 22, 27] [3, 5, 8, 11, 23, 28, 32] [6, 9, 12, 14, 18, 20, 24] [7, 10, 13, 15, 21, 25, 30] [0, 1, 8, 11, 26, 31, 34] [6, 9, 12, 14, 19, 22, 27] [7, 10, 13, 15, 23, 28, 32] [0, 1, 8, 11, 29, 33, 35] [12, 14, 16, 17, 21, 25, 30] [2, 4, 13, 15, 26, 31, 34] [12, 14, 16, 17, 23, 28, 32] [2, 4, 13, 15, 29, 33, 35] [6, 9, 16, 17, 26, 31, 34] [6, 9, 16, 17, 29, 33, 35]
Code ID 36-16-2 · download JSON · raw on GitHub