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[[42,4,7]] d =
n
42
k
4
d
7
kd²/n
4.667
w
7
X/Z
1
g
0.0034
r
6.0828
layers
2
swaps
124

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Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[1, 8, 10, 12, 24, 37, 40]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 1, 5, 7, 10, 21, 25]
certificate exact, d = 7 · scipy/HiGHS MILP
X: no logical < 7 exists; Z: no logical < 7 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 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×21 (2,2)×21 (3,2)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 21 (1,5): 21 (2,2): 21 (2,3): 21 (2,5): 168 (2,6): 42 (2,8): 126 (3,2): 21 (3,3): 154 (3,4): 21 (3,5): 693 (3,6): 840 (3,7): 252 (3,8): 1449 (3,9): 567 (3,10): 168 (3,11): 378 (3,13): 42
trapping sets H_Z (1,2)×21 (2,2)×21 (3,2)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 21 (1,5): 21 (2,2): 21 (2,3): 21 (2,5): 168 (2,6): 42 (2,8): 126 (3,2): 21 (3,3): 154 (3,4): 21 (3,5): 693 (3,6): 840 (3,7): 252 (3,8): 1449 (3,9): 567 (3,10): 168 (3,11): 378 (3,13): 42
witness diameter X 5.3852 · Z 6.0 (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 = 6.083
X checkZ checkqubit site (21)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 124 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(21,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL2_order21_k4.txt, line 1. nonidentity supports a=[3], b=[2, 3, 7, 9].
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

[[42,4,7]] — two-block group-algebra code on SmallGroup(21,2)

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 = 4.67.

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_wt7wtL2_order21_k4.txt, line 1. This row is SmallGroup(21,2) with nonidentity GAP supports a=[3], b=[2, 3, 7, 9]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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(21,2), 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=[3], b=[2, 3, 7, 9] 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 = 42 - rank(HX) - rank(HZ) = 4. 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 21 (max weight 7) · Z-checks 21 (max weight 7)
H_X (21 checks, sparse supports)
[0, 17, 21, 24, 32, 38, 40] [1, 19, 21, 22, 34, 40, 41] [0, 2, 21, 22, 23, 27, 35] [3, 20, 22, 24, 36, 38, 41] [1, 4, 22, 23, 24, 25, 37] [2, 5, 23, 25, 26, 30, 38] [3, 6, 21, 24, 25, 27, 39] [4, 7, 25, 26, 27, 28, 40] [5, 8, 21, 26, 28, 29, 33] [6, 9, 23, 27, 28, 30, 41] [7, 10, 22, 28, 29, 30, 31] [8, 11, 23, 29, 31, 32, 36] [9, 12, 24, 26, 30, 31, 33] [10, 13, 25, 31, 32, 33, 34] [11, 14, 26, 32, 34, 35, 39] [12, 15, 27, 29, 33, 34, 36] [13, 16, 28, 34, 35, 36, 37] [14, 17, 29, 35, 37, 38, 41] [15, 18, 30, 32, 36, 37, 39] [16, 19, 31, 37, 38, 39, 40] [18, 20, 33, 35, 39, 40, 41]
H_Z (21 checks, sparse supports)
[0, 1, 2, 6, 8, 21, 23] [1, 2, 3, 4, 10, 22, 25] [2, 4, 5, 9, 11, 23, 26] [0, 3, 4, 6, 12, 24, 27] [4, 5, 6, 7, 13, 25, 28] [5, 7, 8, 12, 14, 26, 29] [2, 6, 7, 9, 15, 27, 30] [7, 8, 9, 10, 16, 28, 31] [8, 10, 11, 15, 17, 29, 32] [5, 9, 10, 12, 18, 30, 33] [10, 11, 12, 13, 19, 31, 34] [0, 11, 13, 14, 18, 32, 35] [8, 12, 13, 15, 20, 33, 36] [1, 13, 14, 15, 16, 34, 37] [2, 14, 16, 17, 20, 35, 38] [3, 11, 15, 16, 18, 36, 39] [4, 16, 17, 18, 19, 37, 40] [0, 3, 5, 17, 19, 21, 38] [6, 14, 18, 19, 20, 39, 41] [0, 1, 7, 19, 20, 22, 40] [1, 3, 9, 17, 20, 24, 41]
Code ID 42-4-7 · download JSON · raw on GitHub