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[[62,10,7]] d ≤
n
62
k
10
d
7
kd²/n
7.903
w
7
X/Z
1
g
0.0188
r
6.4031
layers
1
swaps
418

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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)
[8, 14, 29, 39, 45, 49, 60]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[1, 13, 14, 19, 31, 50, 56]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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)×31 (2,3)×62 (3,3)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 31 (1,4): 31 (2,3): 62 (2,4): 124 (2,5): 248 (2,6): 124 (3,3): 62 (3,4): 589 (3,5): 1240 (3,6): 1984 (3,7): 2046 (3,8): 682 (3,9): 186 (3,10): 31
trapping sets H_Z (1,3)×31 (2,3)×62 (3,3)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 31 (1,4): 31 (2,3): 62 (2,4): 124 (2,5): 248 (2,6): 124 (3,3): 62 (3,4): 589 (3,5): 1240 (3,6): 1984 (3,7): 2046 (3,8): 682 (3,9): 186 (3,10): 31
witness diameter X 8.9443 · Z 5.831 (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 = 6.403
X checkZ checkqubit site (62)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 418 nearest-neighbor SWAPs per round in total, at most 11 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 contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[62,10,7]] — two-block group-algebra code on SmallGroup(31,1)

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 = 7.90.

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_order31_k10.txt, line 1. This row is SmallGroup(31,1) with nonidentity GAP supports a=[2, 13], b=[2, 3, 28]. 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 ≤ 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

Union Alpha (maker currently anonymous), 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(31,1), 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, 13] and b=[2, 3, 28] 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 = 62 - 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 31 (max weight 7) · Z-checks 31 (max weight 7)
H_X (31 checks, sparse supports)
[0, 19, 30, 31, 35, 60, 61] [0, 1, 20, 31, 32, 36, 61] [1, 2, 21, 31, 32, 33, 37] [2, 3, 22, 32, 33, 34, 38] [3, 4, 23, 33, 34, 35, 39] [4, 5, 24, 34, 35, 36, 40] [5, 6, 25, 35, 36, 37, 41] [6, 7, 26, 36, 37, 38, 42] [7, 8, 27, 37, 38, 39, 43] [8, 9, 28, 38, 39, 40, 44] [9, 10, 29, 39, 40, 41, 45] [10, 11, 30, 40, 41, 42, 46] [0, 11, 12, 41, 42, 43, 47] [1, 12, 13, 42, 43, 44, 48] [2, 13, 14, 43, 44, 45, 49] [3, 14, 15, 44, 45, 46, 50] [4, 15, 16, 45, 46, 47, 51] [5, 16, 17, 46, 47, 48, 52] [6, 17, 18, 47, 48, 49, 53] [7, 18, 19, 48, 49, 50, 54] [8, 19, 20, 49, 50, 51, 55] [9, 20, 21, 50, 51, 52, 56] [10, 21, 22, 51, 52, 53, 57] [11, 22, 23, 52, 53, 54, 58] [12, 23, 24, 53, 54, 55, 59] [13, 24, 25, 54, 55, 56, 60] [14, 25, 26, 55, 56, 57, 61] [15, 26, 27, 31, 56, 57, 58] [16, 27, 28, 32, 57, 58, 59] [17, 28, 29, 33, 58, 59, 60] [18, 29, 30, 34, 59, 60, 61]
H_Z (31 checks, sparse supports)
[0, 1, 2, 27, 31, 32, 43] [1, 2, 3, 28, 32, 33, 44] [2, 3, 4, 29, 33, 34, 45] [3, 4, 5, 30, 34, 35, 46] [0, 4, 5, 6, 35, 36, 47] [1, 5, 6, 7, 36, 37, 48] [2, 6, 7, 8, 37, 38, 49] [3, 7, 8, 9, 38, 39, 50] [4, 8, 9, 10, 39, 40, 51] [5, 9, 10, 11, 40, 41, 52] [6, 10, 11, 12, 41, 42, 53] [7, 11, 12, 13, 42, 43, 54] [8, 12, 13, 14, 43, 44, 55] [9, 13, 14, 15, 44, 45, 56] [10, 14, 15, 16, 45, 46, 57] [11, 15, 16, 17, 46, 47, 58] [12, 16, 17, 18, 47, 48, 59] [13, 17, 18, 19, 48, 49, 60] [14, 18, 19, 20, 49, 50, 61] [15, 19, 20, 21, 31, 50, 51] [16, 20, 21, 22, 32, 51, 52] [17, 21, 22, 23, 33, 52, 53] [18, 22, 23, 24, 34, 53, 54] [19, 23, 24, 25, 35, 54, 55] [20, 24, 25, 26, 36, 55, 56] [21, 25, 26, 27, 37, 56, 57] [22, 26, 27, 28, 38, 57, 58] [23, 27, 28, 29, 39, 58, 59] [24, 28, 29, 30, 40, 59, 60] [0, 25, 29, 30, 41, 60, 61] [0, 1, 26, 30, 31, 42, 61]
Code ID 62-10-7 · download JSON · raw on GitHub