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[[18,8,2]] d =
n
18
k
8
d
2
kd²/n
1.778
w
7
X/Z
1
g
0.0278
r
2.8284
layers
2
swaps
14

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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)
[14, 16]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[2, 4]
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)×9 (2,0)×9 (3,0)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 9 (1,4): 9 (2,0): 9 (2,3): 27 (2,4): 18 (2,5): 54 (2,6): 18 (3,0): 6 (3,3): 111 (3,4): 279 (3,5): 189 (3,6): 123 (3,9): 27
trapping sets H_Z (1,3)×9 (2,0)×9 (3,0)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 9 (1,4): 9 (2,0): 9 (2,3): 27 (2,4): 18 (2,5): 54 (2,6): 18 (3,0): 6 (3,3): 111 (3,4): 279 (3,5): 189 (3,6): 123 (3,9): 27
witness diameter X 2.2361 · Z 1.4142 (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 = 2.828
X checkZ checkqubit site (9)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 14 nearest-neighbor SWAPs per round in total, at most 2 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(9,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order9_k8.txt, line 1. nonidentity supports a=[2, 4], b=[2, 3, 8].
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

[[18,8,2]] — two-block group-algebra code on SmallGroup(9,2)

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_order9_k8.txt, line 1. This row is SmallGroup(9,2) with nonidentity GAP supports a=[2, 4], b=[2, 3, 8]. 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 ≤ 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(9,2), 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=[2, 4], b=[2, 3, 8] 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 = 18 - 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 9 (max weight 7) · Z-checks 9 (max weight 7)
H_X (9 checks, sparse supports)
[0, 1, 3, 9, 12, 14, 15] [0, 1, 3, 9, 10, 11, 16] [2, 4, 6, 9, 11, 15, 17] [0, 1, 3, 10, 12, 13, 17] [2, 4, 6, 10, 11, 13, 14] [5, 7, 8, 11, 12, 14, 17] [2, 4, 6, 12, 13, 15, 16] [5, 7, 8, 9, 13, 14, 16] [5, 7, 8, 10, 15, 16, 17]
H_Z (9 checks, sparse supports)
[0, 1, 2, 7, 9, 10, 12] [1, 3, 4, 8, 9, 10, 12] [1, 2, 4, 5, 11, 13, 15] [0, 3, 5, 6, 9, 10, 12] [3, 4, 6, 7, 11, 13, 15] [0, 4, 5, 7, 14, 16, 17] [0, 2, 6, 8, 11, 13, 15] [1, 6, 7, 8, 14, 16, 17] [2, 3, 5, 8, 14, 16, 17]
Code ID 18-8-2 · download JSON · raw on GitHub