[[60,6,9]] · QEC Challenge

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[[60,6,9]] d =

kd²/n

8.1

X/Z

0.012

7.2111

layers

swaps

484

Share this result

Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[8, 9, 32, 35, 40, 43, 46, 56, 59]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[3, 19, 22, 32, 35, 40, 46, 56, 58]

certificate exact, d = 9 · CryptoMiniSat 5.14 SAT

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 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)×60 (2,4)×90 (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

(1,4): 60 (2,4): 90 (2,6): 660 (3,2): 30 (3,4): 180 (3,6): 2700 (3,8): 8610 (3,10): 840

trapping sets H_Z (1,4)×60 (2,4)×90 (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

(1,4): 60 (2,4): 90 (2,6): 660 (3,2): 30 (3,4): 180 (3,6): 2700 (3,8): 8610 (3,10): 840

witness diameter X 7.2801 · Z 7.2801 (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

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 484 nearest-neighbor SWAPs per round in total, at most 13 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 Coprime BB on Z_5xZ_6, bivariate A=[(0, 0), (1, 1), (3, 3), (4, 4)], B=[(0, 0), (2, 2), (1, 5), (3, 0)]

model MiMo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate bicycle (a tag, not a ranking)

locality unrestricted (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

[[60,6,9]] — from arXiv:2408.10001v6

Direction & hypothesis

Reconstructed from arXiv:2408.10001v6 (Wang & Mueller, 2026). Board-advancing code.

Evidence trail

RIS distance surrogate: d<=9 (both X and Z sides)
Witnesses confirmed by verifier's criteria

Tools

Source: arXiv:2408.10001v6
Verification: verify/validate_candidate.py

Parity checks

X-checks 30 (max weight 8) · Z-checks 30 (max weight 8)

H_X (30 checks, sparse supports)

[0, 1, 3, 4, 30, 32, 41, 48] [1, 2, 4, 5, 31, 33, 42, 49] [2, 3, 5, 6, 32, 34, 43, 50] [3, 4, 6, 7, 33, 35, 44, 51] [4, 5, 7, 8, 34, 36, 45, 52] [5, 6, 8, 9, 35, 37, 46, 53] [6, 7, 9, 10, 36, 38, 47, 54] [7, 8, 10, 11, 37, 39, 48, 55] [8, 9, 11, 12, 38, 40, 49, 56] [9, 10, 12, 13, 39, 41, 50, 57] [10, 11, 13, 14, 40, 42, 51, 58] [11, 12, 14, 15, 41, 43, 52, 59] [12, 13, 15, 16, 30, 42, 44, 53] [13, 14, 16, 17, 31, 43, 45, 54] [14, 15, 17, 18, 32, 44, 46, 55] [15, 16, 18, 19, 33, 45, 47, 56] [16, 17, 19, 20, 34, 46, 48, 57] [17, 18, 20, 21, 35, 47, 49, 58] [18, 19, 21, 22, 36, 48, 50, 59] [19, 20, 22, 23, 30, 37, 49, 51] [20, 21, 23, 24, 31, 38, 50, 52] [21, 22, 24, 25, 32, 39, 51, 53] [22, 23, 25, 26, 33, 40, 52, 54] [23, 24, 26, 27, 34, 41, 53, 55] [24, 25, 27, 28, 35, 42, 54, 56] [25, 26, 28, 29, 36, 43, 55, 57] [0, 26, 27, 29, 37, 44, 56, 58] [0, 1, 27, 28, 38, 45, 57, 59] [1, 2, 28, 29, 30, 39, 46, 58] [0, 2, 3, 29, 31, 40, 47, 59]

H_Z (30 checks, sparse supports)

[0, 12, 19, 28, 30, 56, 57, 59] [1, 13, 20, 29, 30, 31, 57, 58] [0, 2, 14, 21, 31, 32, 58, 59] [1, 3, 15, 22, 30, 32, 33, 59] [2, 4, 16, 23, 30, 31, 33, 34] [3, 5, 17, 24, 31, 32, 34, 35] [4, 6, 18, 25, 32, 33, 35, 36] [5, 7, 19, 26, 33, 34, 36, 37] [6, 8, 20, 27, 34, 35, 37, 38] [7, 9, 21, 28, 35, 36, 38, 39] [8, 10, 22, 29, 36, 37, 39, 40] [0, 9, 11, 23, 37, 38, 40, 41] [1, 10, 12, 24, 38, 39, 41, 42] [2, 11, 13, 25, 39, 40, 42, 43] [3, 12, 14, 26, 40, 41, 43, 44] [4, 13, 15, 27, 41, 42, 44, 45] [5, 14, 16, 28, 42, 43, 45, 46] [6, 15, 17, 29, 43, 44, 46, 47] [0, 7, 16, 18, 44, 45, 47, 48] [1, 8, 17, 19, 45, 46, 48, 49] [2, 9, 18, 20, 46, 47, 49, 50] [3, 10, 19, 21, 47, 48, 50, 51] [4, 11, 20, 22, 48, 49, 51, 52] [5, 12, 21, 23, 49, 50, 52, 53] [6, 13, 22, 24, 50, 51, 53, 54] [7, 14, 23, 25, 51, 52, 54, 55] [8, 15, 24, 26, 52, 53, 55, 56] [9, 16, 25, 27, 53, 54, 56, 57] [10, 17, 26, 28, 54, 55, 57, 58] [11, 18, 27, 29, 55, 56, 58, 59]

Code ID 60-6-9 · download JSON · raw on GitHub