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[[48,8,6]] d =
n
48
k
8
d
6
kd²/n
6.0
w
7
X/Z
1
g
0.0089
r
5.099
layers
2
swaps
157

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 9, 29, 31, 40, 42]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[7, 10, 13, 16, 45, 47]
certificate exact, d = 6 · scipy/HiGHS MILP
X: no logical < 6 exists; Z: no logical < 6 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)×24 (2,4)×84 (3,3)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 24 (1,4): 24 (2,4): 84 (2,5): 288 (2,6): 120 (3,3): 24 (3,4): 216 (3,5): 744 (3,6): 2072 (3,7): 2256 (3,8): 936 (3,9): 360 (3,10): 48
trapping sets H_Z (1,3)×24 (2,4)×84 (3,3)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 24 (1,4): 24 (2,4): 84 (2,5): 288 (2,6): 120 (3,3): 24 (3,4): 216 (3,5): 744 (3,6): 2072 (3,7): 2256 (3,8): 936 (3,9): 360 (3,10): 48
witness diameter X 3.6056 · Z 3.1623 (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 = 5.099
X checkZ checkqubit site (24)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 157 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(24,9), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order24_k8.txt, line 2. nonidentity supports a=[7, 17], b=[2, 4, 18].
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

[[48,8,6]] — two-block group-algebra code on SmallGroup(24,9)

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 = 6.00.

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_wt7wtL3_order24_k8.txt, line 2. This row is SmallGroup(24,9) with nonidentity GAP supports a=[7, 17], b=[2, 4, 18]. 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 ≤ 6, not an exact certificate. The published distance (d=6) 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(24,9), 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=[7, 17], b=[2, 4, 18] 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 = 48 - 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 24 (max weight 7) · Z-checks 24 (max weight 7)
H_X (24 checks, sparse supports)
[0, 8, 21, 24, 31, 34, 46] [1, 10, 12, 24, 25, 38, 47] [2, 3, 23, 26, 37, 40, 42] [3, 7, 16, 24, 27, 33, 39] [4, 14, 17, 25, 28, 40, 42] [5, 6, 16, 26, 29, 43, 45] [0, 6, 19, 25, 27, 30, 37] [7, 18, 20, 28, 31, 43, 45] [8, 10, 13, 26, 28, 32, 44] [9, 11, 19, 29, 33, 34, 46] [2, 10, 15, 27, 34, 41, 45] [1, 11, 22, 26, 28, 30, 35] [2, 12, 14, 29, 31, 32, 36] [13, 15, 22, 33, 37, 38, 47] [3, 5, 14, 30, 34, 38, 44] [4, 15, 23, 29, 31, 35, 39] [0, 16, 20, 32, 35, 40, 47] [5, 17, 18, 24, 33, 36, 41] [6, 9, 18, 32, 35, 38, 42] [1, 8, 19, 36, 39, 40, 43] [9, 20, 21, 25, 37, 41, 44] [11, 13, 21, 36, 39, 42, 45] [4, 12, 22, 27, 41, 43, 46] [7, 17, 23, 30, 44, 46, 47]
H_Z (24 checks, sparse supports)
[0, 1, 3, 17, 24, 30, 40] [1, 4, 6, 20, 25, 35, 43] [2, 5, 8, 11, 26, 34, 36] [3, 6, 10, 22, 26, 27, 38] [4, 7, 8, 11, 28, 39, 46] [5, 9, 12, 15, 29, 38, 41] [6, 11, 14, 23, 29, 30, 42] [0, 7, 12, 15, 27, 31, 47] [8, 12, 16, 18, 24, 32, 43] [3, 9, 13, 17, 33, 42, 44] [0, 9, 10, 14, 25, 32, 34] [11, 15, 16, 18, 33, 35, 45] [12, 17, 19, 21, 25, 36, 46] [2, 6, 13, 20, 32, 37, 45] [1, 13, 14, 18, 28, 36, 38] [3, 15, 19, 21, 34, 37, 39] [2, 4, 16, 19, 27, 29, 40] [10, 17, 20, 22, 28, 41, 47] [2, 4, 18, 21, 31, 41, 42] [5, 7, 19, 22, 30, 33, 43] [8, 14, 20, 23, 31, 40, 44] [5, 7, 10, 21, 24, 44, 45] [0, 9, 22, 23, 35, 37, 46] [1, 13, 16, 23, 26, 39, 47]
Code ID 48-8-6 · download JSON · raw on GitHub