[[66,5,5]] · QEC Challenge

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[[66,5,5]] d =

kd²/n

1.894

0.0189

3.1623

layers

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Distance

d_X 5 · witness weight 5 (claimed exact)

witness operator (support, 5 qubits)

[2, 26, 37, 50, 57]

d_Z 5 · witness weight 5 (claimed exact)

witness operator (support, 5 qubits)

[24, 28, 57, 58, 61]

certificate exact, d = 5 · scipy/HiGHS MILP

X: no logical < 5 exists; Z: no logical < 5 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (36)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

Construction & provenance

authors @FarLab and @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Generalized weight-6, (6,6) open-boundary lattice. New instance of the Liang-Eberhardt-Chen open-boundary family (arXiv:2504.08887); the paper has no k=5 family.

model Claude Opus 4.8 (claimed, not verified)

builds on arXiv:2504.08887

date 2026-06-19

notes Distance proven exact by cutoff MILP (scipy/HiGHS); no lighter logical exists on either side. 2D-local bilayer layout (reconstructed 2026-07-02): A/B sublattices of 6x6 and 5x6 rows on one grid, physical row = (+i+1) mod 6 on A and (+i+0) mod 5 on B; interaction radius 3.162.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 29 · Z-checks 33

H_X (29 checks, sparse supports)

[0, 7, 13, 37, 38, 42] [1, 8, 14, 38, 39, 43] [2, 9, 15, 39, 40, 44] [3, 10, 16, 40, 41, 45] [6, 13, 19, 43, 44, 48] [7, 14, 20, 44, 45, 49] [8, 15, 21, 45, 46, 50] [9, 16, 22, 46, 47, 51] [12, 19, 25, 49, 50, 54] [13, 20, 26, 50, 51, 55] [14, 21, 27, 51, 52, 56] [15, 22, 28, 52, 53, 57] [1, 30] [2, 31] [3, 32] [4, 33] [5, 34] [24, 61, 62] [25, 62, 63] [26, 63, 64] [27, 64, 65] [7, 30, 31, 32, 36] [8, 31, 32, 33, 37] [9, 32, 33, 34, 38] [10, 33, 34, 35, 39] [18, 25, 55, 56, 60] [19, 26, 56, 57, 61] [20, 27, 57, 58, 62] [21, 28, 58, 59, 63]

H_Z (33 checks, sparse supports)

[2, 6, 7, 31, 37, 44] [3, 7, 8, 32, 38, 45] [4, 8, 9, 33, 39, 46] [5, 9, 10, 34, 40, 47] [8, 12, 13, 37, 43, 50] [9, 13, 14, 38, 44, 51] [10, 14, 15, 39, 45, 52] [11, 15, 16, 40, 46, 53] [14, 18, 19, 43, 49, 56] [15, 19, 20, 44, 50, 57] [16, 20, 21, 45, 51, 58] [17, 21, 22, 46, 52, 59] [20, 24, 25, 49, 55, 62] [21, 25, 26, 50, 56, 63] [22, 26, 27, 51, 57, 64] [23, 27, 28, 52, 58, 65] [11] [17] [23] [29] [0, 42] [6, 48] [12, 54] [18, 60] [10, 35, 41] [16, 41, 47] [22, 47, 53] [28, 53, 59] [1, 30, 36, 43, 48] [7, 36, 42, 49, 54] [13, 42, 48, 55, 60] [0, 6, 13, 18, 55] [6, 12, 19, 24, 61]

raw submission JSON