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[[64,22,4]] d =
n
64
k
22
d
4
kd²/n
5.5
w
8
X/Z
1
g
0.0065
r
5.3852
layers
2
swaps
241

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[0, 9, 33, 35]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[31, 54, 60, 63]
certificate exact, d = 4 · scipy/HiGHS MILP
X: no logical < 4 exists; Z: no logical < 4 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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×64 (2,4)×256 (3,4)×640 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 64 (2,4): 256 (2,6): 384 (3,4): 640 (3,6): 3840 (3,8): 4352
trapping sets H_Z (1,4)×64 (2,4)×256 (3,4)×640 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 64 (2,4): 256 (2,6): 384 (3,4): 640 (3,6): 3840 (3,8): 4352
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 @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-02
r = 5.385
X checkZ checkqubit site (32)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 241 nearest-neighbor SWAPs per round in total, at most 7 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(32,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL4_order32_k22.txt, line 1. nonidentity supports a=[2, 3, 7], b=[3, 8, 17].
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

[[64,22,4]] — two-block group-algebra code on SmallGroup(32,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 5.50.

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_wt8wtL4_order32_k22.txt, line 1. This row is SmallGroup(32,3) with nonidentity GAP supports a=[2, 3, 7], b=[3, 8, 17]. 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=22), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance 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(32,3), 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, 3, 7], b=[3, 8, 17] 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 = 64 - rank(HX) - rank(HZ) = 22. 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 32 (max weight 8) · Z-checks 32 (max weight 8)
H_X (32 checks, sparse supports)
[0, 11, 20, 31, 32, 41, 43, 60] [0, 1, 11, 17, 33, 46, 49, 62] [0, 2, 20, 27, 32, 34, 41, 50] [1, 3, 17, 22, 35, 52, 54, 63] [2, 4, 27, 29, 34, 36, 50, 53] [5, 7, 24, 26, 33, 37, 49, 56] [0, 1, 2, 6, 33, 38, 46, 55] [3, 7, 22, 26, 32, 39, 43, 58] [2, 4, 6, 8, 38, 40, 55, 57] [5, 9, 24, 28, 35, 41, 54, 60] [1, 3, 6, 10, 35, 42, 52, 59] [4, 11, 29, 31, 36, 43, 53, 60] [5, 7, 12, 16, 33, 37, 38, 44] [6, 8, 10, 13, 42, 45, 59, 61] [9, 14, 28, 30, 39, 46, 58, 62] [12, 15, 16, 19, 38, 40, 44, 47] [3, 7, 10, 16, 32, 34, 39, 48] [4, 8, 11, 17, 40, 49, 57, 62] [5, 9, 12, 18, 35, 41, 42, 50] [10, 13, 16, 19, 34, 36, 48, 51] [14, 20, 30, 31, 37, 52, 56, 63] [12, 15, 18, 21, 42, 45, 50, 53] [8, 13, 17, 22, 45, 54, 61, 63] [9, 14, 18, 23, 39, 46, 48, 55] [15, 19, 24, 26, 40, 47, 49, 56] [18, 21, 23, 25, 48, 51, 55, 57] [13, 19, 22, 26, 36, 43, 51, 58] [14, 20, 23, 27, 37, 44, 52, 59] [15, 21, 24, 28, 45, 53, 54, 60] [23, 25, 27, 29, 44, 47, 59, 61] [21, 25, 28, 30, 51, 57, 58, 62] [25, 29, 30, 31, 47, 56, 61, 63]
H_Z (32 checks, sparse supports)
[0, 2, 7, 16, 32, 33, 34, 38] [1, 5, 6, 12, 33, 35, 38, 42] [2, 4, 16, 19, 34, 36, 38, 40] [3, 9, 10, 18, 35, 39, 42, 48] [4, 11, 19, 26, 36, 40, 43, 49] [5, 12, 20, 27, 37, 41, 44, 50] [6, 8, 12, 15, 38, 40, 42, 45] [7, 14, 16, 23, 37, 39, 44, 48] [8, 15, 17, 24, 40, 45, 49, 54] [0, 2, 9, 18, 41, 46, 50, 55] [10, 13, 18, 21, 42, 45, 48, 51] [0, 7, 11, 26, 32, 33, 43, 49] [12, 15, 27, 29, 44, 47, 50, 53] [13, 21, 22, 28, 45, 51, 54, 58] [1, 6, 14, 23, 46, 52, 55, 59] [15, 24, 29, 31, 47, 53, 56, 60] [16, 19, 23, 25, 44, 47, 48, 51] [1, 5, 17, 24, 33, 35, 49, 54] [2, 4, 18, 21, 50, 53, 55, 57] [19, 25, 26, 30, 47, 51, 56, 58] [3, 10, 20, 27, 32, 34, 52, 59] [4, 11, 21, 28, 53, 57, 60, 62] [3, 9, 22, 28, 35, 39, 54, 58] [6, 8, 23, 25, 55, 57, 59, 61] [5, 20, 24, 31, 37, 41, 56, 60] [8, 17, 25, 30, 57, 61, 62, 63] [7, 14, 26, 30, 37, 39, 56, 58] [10, 13, 27, 29, 34, 36, 59, 61] [0, 9, 11, 28, 41, 46, 60, 62] [13, 22, 29, 31, 36, 43, 61, 63] [1, 14, 17, 30, 46, 52, 62, 63] [3, 20, 22, 31, 32, 43, 52, 63]
Code ID 64-22-4 · download JSON · raw on GitHub