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[[42,10,5]] d =
n
42
k
10
d
5
kd²/n
5.952
w
7
X/Z
1
g
0.0043
r
6.0828
layers
2
swaps
124

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[0, 9, 10, 37, 38]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[13, 14, 21, 27, 31]
certificate exact, d = 5 · scipy/HiGHS MILP
X: no logical < 5 exists; Z: no logical < 5 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)×21 (2,3)×21 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 21 (1,4): 21 (2,3): 21 (2,4): 63 (2,5): 210 (2,6): 126 (3,3): 63 (3,4): 336 (3,5): 588 (3,6): 1428 (3,7): 1869 (3,8): 609 (3,9): 252 (3,10): 84
trapping sets H_Z (1,3)×21 (2,3)×21 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 21 (1,4): 21 (2,3): 21 (2,4): 63 (2,5): 210 (2,6): 126 (3,3): 63 (3,4): 336 (3,5): 588 (3,6): 1428 (3,7): 1869 (3,8): 609 (3,9): 252 (3,10): 84
witness diameter X 4.4721 · 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 @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_wt7wtL3_order21_k10.txt, line 1. nonidentity supports a=[5, 13], b=[3, 10, 19].
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

[[42,10,5]] — 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 = 5.95.

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_order21_k10.txt, line 1. This row is SmallGroup(21,2) with nonidentity GAP supports a=[5, 13], b=[3, 10, 19]. 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=10), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 5, not an exact certificate. The published distance (d=5) 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=[5, 13], b=[3, 10, 19] 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) = 10. 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, 13, 20, 21, 28, 37, 38] [1, 15, 17, 22, 30, 39, 40] [2, 3, 16, 21, 23, 31, 40] [3, 11, 19, 24, 26, 35, 41] [0, 4, 18, 22, 25, 33, 41] [5, 6, 19, 22, 23, 26, 34] [1, 6, 14, 24, 27, 29, 38] [2, 7, 20, 24, 25, 28, 36] [1, 8, 9, 25, 26, 29, 37] [4, 9, 17, 21, 27, 30, 32] [3, 5, 10, 27, 28, 31, 39] [4, 11, 12, 28, 29, 32, 40] [0, 7, 12, 23, 30, 33, 35] [6, 8, 13, 30, 31, 34, 41] [7, 14, 15, 22, 31, 32, 35] [2, 10, 15, 26, 33, 36, 38] [9, 11, 16, 24, 33, 34, 37] [10, 17, 18, 25, 34, 35, 38] [5, 13, 18, 21, 29, 36, 39] [12, 14, 19, 27, 36, 37, 40] [8, 16, 20, 23, 32, 39, 41]
H_Z (21 checks, sparse supports)
[0, 2, 9, 18, 21, 25, 33] [1, 4, 5, 14, 22, 27, 29] [2, 5, 12, 20, 23, 28, 36] [3, 6, 7, 16, 23, 24, 31] [4, 7, 8, 17, 25, 30, 32] [3, 5, 8, 15, 26, 31, 39] [6, 9, 10, 19, 26, 27, 34] [0, 7, 10, 11, 28, 33, 35] [6, 8, 11, 18, 29, 34, 41] [1, 9, 12, 13, 29, 30, 37] [2, 10, 13, 14, 31, 36, 38] [9, 11, 14, 20, 24, 32, 37] [4, 12, 15, 16, 32, 33, 40] [5, 13, 16, 17, 21, 34, 39] [3, 12, 14, 17, 27, 35, 40] [7, 15, 18, 19, 22, 35, 36] [0, 8, 16, 19, 23, 37, 41] [0, 6, 15, 17, 22, 30, 38] [1, 10, 18, 20, 25, 38, 39] [1, 2, 11, 19, 24, 26, 40] [3, 4, 13, 20, 21, 28, 41]
Code ID 42-10-5 · download JSON · raw on GitHub