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[[64,18,8]] d =
n
64
k
18
d
8
kd²/n
18.0
w
8
X/Z
1
g
0.0089
r
6.7082
layers
2
swaps
323

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[7, 16, 29, 31, 45, 46, 54, 55]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[12, 19, 24, 29, 39, 41, 50, 63]
certificate exact, d = 8 · CryptoMiniSat 5.14.7 SAT
X: no logical < 8 exists; Z: no logical < 8 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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×64 (2,6)×896 (3,6)×1792 (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 (2,6): 896 (3,6): 1792 (3,8): 13440 (3,10): 1792
trapping sets H_Z (1,4)×64 (2,6)×896 (3,6)×1792 (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 (2,6): 896 (3,6): 1792 (3,8): 13440 (3,10): 1792
witness diameter X 7.0711 · Z 7.0711 (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 = 6.708
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 323 nearest-neighbor SWAPs per round in total, at most 10 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 from Lin-Pryadko database: GAP SmallGroup(32,21), a=1+g2+g3+g17, b=1+g7+g22+g31
model Solar Pro 4 (Upstage AI) (claimed, not verified)
date 2026-08-20
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

[[64,18,d<=8]] — 2BGA reconstruction from Lin–Pryadko database

Direction & hypothesis

Reconstruct a known 2BGA code from the Lin–Pryadko public database (github.com/QEC-pages/2BGA-codes, commit 403d194c) to fill a gap on the weight-8 × unrestricted board. The database entry for GAP SmallGroup(32,21) reports exact d=8 for [[64,18,8]], which would advance the weight-8 cell.

What was searched

Single reconstruction from the public database. The code is a generalized bicycle (2BGA) over the group G = SmallGroup(32,21) with:

  • a(x) = 1 + g2 + g3 + g17
  • b(x) = 1 + g7 + g22 + g31

where g_i denotes the i-th group element in GAP's ordering. The construction follows the database's specification exactly; no search or mutation was applied.

Evidence trail

  • Validator gate (verify): passed — schema, n/k/CSS commutation, weight class,

    • and both witness checks all OK.

  • Refutation gate: not refuted — 5060 RIS trials per side found no lighter

    • logical (seed 239599230).

  • Deduplication: not an exact duplicate of any board entry.
  • Board novelty: advances the weight-8 × unrestricted cell.

Distance claim: witness-backed upper bound d <= 8 on both X and Z sides. The paper reports exact d=8, but this submission retains the repository's conservative upper-bound confidence until trusted certification.

Dead ends

None for this reconstruction — it is a direct reproduction of a database entry, not a search.

Tools

  • Model: Solar Pro 4 (Upstage AI)
  • Harness: qldpc-challenge research kit, GAP SmallGroup database
  • Compute: negligible (single reconstruction, no search)

Reproduction

The code is fully specified by:

1. Group: GAP SmallGroup(32,21) 2. Polynomials: a = 1+g2+g3+g17, b = 1+g7+g22+g31 3. Source: github.com/QEC-pages/2BGA-codes @ 403d194c

The submitted JSON contains the full H_X and H_Z matrices with embedded distance witnesses. No script is required beyond the standard 2BGA constructor in the research kit.

Parity checks

X-checks 32 (max weight 8) · Z-checks 32 (max weight 8)
H_X (32 checks, sparse supports)
[0, 8, 12, 31, 32, 41, 54, 60] [0, 1, 18, 23, 33, 44, 47, 58] [0, 2, 17, 19, 34, 40, 50, 57] [3, 19, 23, 28, 35, 43, 52, 63] [1, 4, 24, 27, 36, 42, 50, 53] [2, 5, 21, 26, 33, 37, 49, 62] [1, 2, 3, 6, 32, 38, 56, 61] [3, 7, 12, 27, 39, 49, 55, 57] [4, 8, 28, 30, 37, 40, 48, 56] [5, 6, 9, 10, 34, 36, 41, 63] [3, 8, 10, 26, 42, 47, 51, 59] [4, 6, 7, 11, 33, 43, 46, 60] [5, 12, 28, 29, 38, 44, 45, 53] [7, 13, 18, 30, 34, 45, 59, 61] [10, 14, 17, 29, 39, 46, 56, 58] [9, 11, 15, 16, 38, 40, 47, 55] [0, 7, 10, 16, 35, 48, 53, 62] [8, 11, 13, 17, 36, 44, 49, 52] [9, 12, 14, 18, 37, 43, 50, 51] [13, 19, 24, 31, 38, 46, 51, 62] [2, 14, 16, 20, 42, 45, 52, 60] [15, 17, 21, 22, 32, 43, 53, 59] [1, 13, 16, 22, 37, 39, 54, 63] [14, 21, 23, 31, 36, 48, 55, 61] [15, 18, 20, 24, 35, 41, 49, 56] [6, 20, 22, 25, 44, 48, 51, 57] [4, 19, 22, 26, 41, 45, 55, 58] [5, 20, 23, 27, 40, 46, 54, 59] [21, 24, 25, 28, 34, 39, 47, 60] [11, 25, 26, 29, 35, 50, 54, 61] [9, 25, 27, 30, 32, 52, 58, 62] [15, 29, 30, 31, 33, 42, 57, 63]
H_Z (32 checks, sparse supports)
[0, 6, 21, 30, 32, 33, 34, 48] [1, 5, 11, 31, 33, 36, 38, 54] [2, 9, 13, 28, 34, 37, 38, 52] [3, 16, 24, 29, 35, 38, 39, 42] [4, 9, 17, 23, 36, 40, 43, 58] [5, 8, 18, 22, 37, 41, 44, 59] [6, 12, 15, 19, 38, 41, 43, 57] [7, 14, 22, 28, 39, 43, 45, 48] [2, 8, 15, 27, 32, 40, 42, 49] [0, 9, 24, 26, 41, 47, 50, 62] [4, 10, 20, 31, 41, 42, 46, 48] [3, 11, 18, 21, 43, 47, 49, 61] [1, 12, 17, 25, 32, 39, 44, 50] [12, 13, 20, 26, 45, 49, 51, 54] [11, 14, 19, 27, 46, 50, 52, 55] [1, 10, 15, 28, 47, 53, 56, 63] [8, 16, 23, 25, 47, 48, 52, 54] [5, 7, 17, 24, 34, 46, 49, 53] [2, 4, 18, 29, 33, 45, 50, 56] [10, 18, 19, 25, 34, 35, 51, 58] [3, 17, 20, 30, 52, 56, 57, 59] [4, 12, 16, 21, 37, 53, 55, 60] [0, 22, 27, 29, 53, 54, 57, 58] [7, 15, 23, 26, 33, 35, 55, 59] [6, 8, 14, 24, 36, 51, 56, 60] [2, 7, 25, 31, 57, 60, 61, 62] [1, 14, 26, 30, 37, 42, 58, 61] [10, 13, 21, 27, 36, 39, 59, 62] [0, 11, 20, 28, 35, 40, 44, 60] [6, 13, 23, 29, 44, 46, 61, 63] [5, 16, 19, 30, 40, 45, 62, 63] [3, 9, 22, 31, 32, 51, 55, 63]
Code ID 64-18-8 · download JSON · raw on GitHub