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[[50,8,5]] d =
n
50
k
8
d
5
kd²/n
4.0
w
7
X/Z
1
g
0.01
r
4.4721
layers
2
swaps
150

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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)
[13, 17, 37, 41, 42]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[0, 2, 10, 25, 26]
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 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×25 (2,2)×25 (3,2)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 25 (1,5): 25 (2,2): 25 (2,4): 25 (2,5): 250 (2,6): 25 (2,8): 125 (3,2): 50 (3,4): 200 (3,5): 1325 (3,6): 575 (3,7): 425 (3,8): 1975 (3,9): 425 (3,10): 250 (3,11): 550 (3,13): 25
trapping sets H_Z (1,2)×25 (2,2)×25 (3,2)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 25 (1,5): 25 (2,2): 25 (2,4): 25 (2,5): 250 (2,6): 25 (2,8): 125 (3,2): 50 (3,4): 200 (3,5): 1325 (3,6): 575 (3,7): 425 (3,8): 1975 (3,9): 425 (3,10): 250 (3,11): 550 (3,13): 25
witness diameter X 4.1231 · Z 3.6056 (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 = 4.472
X checkZ checkqubit site (25)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 150 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(25,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL2_order25_k8.txt, line 1. nonidentity supports a=[3], b=[2, 4, 7, 16].
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

[[50,8,5]] — two-block group-algebra code on SmallGroup(25,1)

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 = 4.00.

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_order25_k8.txt, line 1. This row is SmallGroup(25,1) with nonidentity GAP supports a=[3], b=[2, 4, 7, 16]. 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 ≤ 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(25,1), 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=[3], b=[2, 4, 7, 16] 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 = 50 - 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 25 (max weight 7) · Z-checks 25 (max weight 7)
H_X (25 checks, sparse supports)
[0, 14, 25, 38, 46, 48, 49] [1, 18, 25, 26, 42, 48, 49] [0, 2, 27, 28, 31, 35, 43] [3, 21, 25, 26, 28, 45, 49] [1, 4, 27, 29, 31, 35, 46] [2, 5, 26, 30, 32, 36, 40] [6, 23, 25, 26, 28, 31, 47] [3, 7, 27, 29, 32, 35, 48] [4, 8, 28, 30, 33, 36, 40] [5, 9, 29, 34, 37, 41, 44] [10, 24, 26, 28, 31, 35, 39] [6, 11, 27, 29, 32, 36, 49] [7, 12, 30, 31, 33, 37, 40] [8, 13, 32, 34, 38, 41, 44] [9, 14, 33, 39, 42, 45, 47] [10, 15, 25, 29, 32, 36, 40] [11, 16, 30, 33, 35, 37, 41] [12, 17, 34, 36, 38, 42, 44] [13, 18, 37, 39, 43, 45, 47] [15, 19, 27, 33, 37, 41, 44] [16, 20, 34, 38, 40, 42, 45] [17, 21, 39, 41, 43, 46, 47] [19, 22, 30, 38, 42, 45, 47] [20, 23, 39, 43, 44, 46, 48] [22, 24, 34, 43, 46, 48, 49]
H_Z (25 checks, sparse supports)
[0, 1, 3, 6, 15, 25, 27] [1, 3, 5, 6, 10, 26, 29] [2, 4, 7, 11, 19, 27, 30] [2, 3, 6, 8, 10, 28, 32] [4, 7, 9, 11, 15, 29, 33] [5, 8, 12, 16, 22, 30, 34] [2, 4, 6, 10, 12, 31, 36] [5, 7, 11, 13, 15, 32, 37] [8, 12, 14, 16, 19, 33, 38] [9, 13, 17, 20, 24, 34, 39] [2, 4, 7, 10, 16, 35, 40] [5, 8, 11, 15, 17, 36, 41] [9, 12, 16, 18, 19, 37, 42] [0, 13, 17, 20, 22, 38, 43] [10, 14, 18, 21, 23, 25, 39] [5, 8, 12, 15, 20, 40, 44] [9, 13, 16, 19, 21, 41, 45] [1, 14, 17, 20, 22, 42, 46] [2, 18, 21, 23, 24, 26, 43] [9, 13, 17, 19, 23, 44, 47] [3, 14, 18, 20, 22, 45, 48] [0, 4, 21, 23, 24, 28, 46] [6, 14, 18, 21, 22, 47, 49] [0, 1, 7, 23, 24, 31, 48] [0, 1, 3, 11, 24, 35, 49]
Code ID 50-8-5 · download JSON · raw on GitHub