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[[60,4,9]] d =
n
60
k
4
d
9
kd²/n
5.4
w
6
X/Z
1
g
0.0128
r
6.4031
layers
1
swaps
409

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[11, 12, 29, 41, 45, 47, 57, 58, 59]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[2, 5, 12, 23, 25, 26, 32, 45, 50]
certificate exact, d = 9 · scipy/HiGHS MILP
X: no logical < 9 exists; Z: no logical < 9 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×30 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 30 (1,4): 30 (2,2): 30 (2,4): 240 (2,6): 180 (3,2): 30 (3,4): 1080 (3,6): 2560 (3,8): 1500 (3,10): 120
trapping sets H_Z (1,2)×30 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 30 (1,4): 30 (2,2): 30 (2,4): 240 (2,6): 180 (3,2): 30 (3,4): 1080 (3,6): 2560 (3,8): 1500 (3,10): 120
witness diameter X 7.2111 · Z 7.2111 (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 (60)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 409 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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[60,4,9]] — two-block group-algebra code on SmallGroup(30,4)

Direction & hypothesis

Target: unrestricted weight-6 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 = 5.40.

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_wt6wtL2_order30_k4.txt, line 1. This row is SmallGroup(30,4) with nonidentity GAP supports a=[8], b=[3, 11, 23]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 9, not an exact certificate. The published distance (d=9) 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(30,4), 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=[8] and b=[3, 11, 23] 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 = 60 - rank(HX) - rank(HZ) = 4. 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 30 (max weight 6) · Z-checks 30 (max weight 6)
H_X (30 checks, sparse supports)
[0, 28, 30, 36, 51, 59] [1, 29, 31, 39, 48, 58] [2, 20, 30, 32, 41, 53] [3, 6, 33, 39, 42, 56] [4, 23, 31, 34, 38, 50] [5, 9, 35, 36, 45, 54] [6, 25, 32, 36, 46, 57] [0, 7, 31, 33, 37, 47] [8, 12, 38, 45, 48, 59] [9, 27, 34, 39, 43, 55] [1, 10, 30, 35, 40, 44] [11, 15, 41, 42, 51, 58] [2, 12, 34, 37, 42, 52] [3, 13, 35, 38, 43, 53] [14, 18, 39, 44, 51, 54] [4, 15, 32, 40, 45, 49] [5, 16, 33, 41, 46, 50] [17, 21, 36, 47, 48, 56] [7, 18, 40, 43, 48, 57] [8, 19, 31, 41, 44, 49] [20, 24, 45, 50, 56, 58] [10, 21, 37, 46, 51, 55] [11, 22, 30, 38, 47, 52] [23, 26, 42, 53, 54, 59] [13, 24, 34, 46, 49, 54] [14, 25, 35, 47, 50, 55] [16, 26, 32, 43, 52, 56] [17, 27, 33, 44, 53, 57] [19, 28, 40, 52, 55, 58] [22, 29, 37, 49, 57, 59]
H_Z (30 checks, sparse supports)
[0, 2, 10, 22, 30, 37] [1, 4, 7, 19, 31, 40] [2, 6, 15, 26, 32, 42] [3, 7, 16, 27, 33, 43] [4, 9, 12, 24, 34, 45] [5, 10, 13, 25, 35, 46] [0, 5, 6, 17, 33, 36] [7, 12, 21, 29, 37, 48] [4, 8, 13, 22, 38, 49] [1, 3, 9, 14, 35, 39] [10, 15, 18, 28, 40, 51] [2, 11, 16, 19, 41, 52] [3, 11, 12, 23, 38, 42] [9, 13, 18, 26, 43, 54] [10, 14, 19, 27, 44, 55] [5, 8, 15, 20, 41, 45] [6, 16, 21, 24, 46, 56] [7, 17, 22, 25, 47, 57] [1, 8, 17, 18, 44, 48] [15, 19, 24, 29, 49, 58] [4, 16, 20, 25, 32, 50] [0, 11, 14, 21, 47, 51] [12, 22, 26, 28, 52, 59] [2, 13, 23, 27, 34, 53] [5, 14, 23, 24, 50, 54] [9, 21, 25, 28, 36, 55] [3, 17, 20, 26, 53, 56] [6, 18, 27, 29, 39, 57] [1, 11, 20, 28, 30, 58] [0, 8, 23, 29, 31, 59]
Code ID 60-4-9 · download JSON · raw on GitHub