[[56,14,6]] · QEC Challenge

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[[56,14,6]] d ≤

kd²/n

9.0

X/Z

0.0214

6.4031

layers

swaps

355

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[13, 14, 15, 30, 32, 35]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[5, 19, 30, 33, 44, 47]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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)×56 (2,4)×112 (3,4)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 56 (2,4): 112 (2,6): 560 (3,4): 336 (3,6): 2828 (3,8): 6580 (3,10): 504

trapping sets H_Z (1,4)×56 (2,4)×112 (3,4)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 56 (2,4): 112 (2,6): 560 (3,4): 336 (3,6): 2828 (3,8): 6580 (3,10): 504

witness diameter X 5.3852 · 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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (56)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 355 nearest-neighbor SWAPs per round in total, at most 12 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_4xZ_7, bivariate A=[(0, 0), (1, 1), (0, 4), (1, 2)], B=[(0, 0), (1, 1), (1, 3), (0, 6)]

model MiMo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate 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

[[56,14,6]] — from arXiv:2408.10001v6

Direction & hypothesis

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

Evidence trail

RIS distance surrogate: d<=6 (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 28 (max weight 8) · Z-checks 28 (max weight 8)

H_X (28 checks, sparse supports)

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

H_Z (28 checks, sparse supports)

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

Code ID 56-14-6 · download JSON · raw on GitHub