[[48,6,6]] · QEC Challenge

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[[48,6,6]] d =

kd²/n

4.5

X/Z

0.0067

5.099

layers

swaps

154

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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)

[3, 7, 11, 15, 19, 23]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[25, 29, 33, 37, 41, 45]

certificate exact, d = 6 · scipy/HiGHS MILP

X: no logical < 6 exists; Z: no logical < 6 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)×48 (2,2)×24 (3,2)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 48 (2,2): 24 (2,4): 48 (2,6): 504 (3,2): 24 (3,4): 384 (3,6): 1976 (3,8): 5688 (3,10): 576

trapping sets H_Z (1,4)×48 (2,2)×24 (3,2)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 48 (2,2): 24 (2,4): 48 (2,6): 504 (3,2): 24 (3,4): 384 (3,6): 1976 (3,8): 5688 (3,10): 576

witness diameter X 4.4721 · Z 3.6056 (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 (24)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 154 nearest-neighbor SWAPs per round in total, at most 8 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_3xZ_8, bivariate A=[(0, 0), (1, 1), (2, 2), (0, 3)], B=[(0, 0), (0, 3), (1, 7), (1, 0)]

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

[[48,6,6]] — 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 24 (max weight 8) · Z-checks 24 (max weight 8)

H_X (24 checks, sparse supports)

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

H_Z (24 checks, sparse supports)

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

Code ID 48-6-6 · download JSON · raw on GitHub