[[39,4,7]] · QEC Challenge

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[[39,4,7]] d ≤

kd²/n

5.02564

X/Z

1.14

0.0155

4.2426

layers

swaps

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Distance

X/Z asymmetry 1.14 · d_X ≤ 8, d_Z ≤ 7 · 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)

[0, 1, 8, 9, 11, 14, 24, 28]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness found by @mathysrennela · gf2_fast · found at 2×10⁵ trials · 2026-11-13

witness operator (support, 7 qubits)

[5, 10, 16, 23, 29, 33, 36]

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 7–8 (mean 7.667)

qubit degrees H_X 1–6 (mean 3.487) · H_Z 1–6 (mean 3.538)

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

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

(1,1): 4 (1,2): 15 (1,4): 4 (1,5): 10 (1,6): 6 (2,1): 4 (2,2): 15 (2,3): 9 (2,4): 49 (2,5): 95 (2,6): 79 (2,7): 52 (2,8): 42 (2,9): 23 (2,10): 6 (3,1): 4 (3,2): 32 (3,3): 99 (3,4): 313 (3,5): 599 (3,6): 785 (3,7): 918 (3,8): 789 (3,9): 521 (3,10): 244 (3,11): 109 (3,12): 52 (3,13): 11 (3,14): 1

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

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

(1,1): 4 (1,2): 16 (1,4): 2 (1,5): 8 (1,6): 9 (2,1): 4 (2,2): 15 (2,3): 6 (2,4): 30 (2,5): 87 (2,6): 92 (2,7): 41 (2,8): 41 (2,9): 32 (2,10): 13 (3,1): 4 (3,2): 21 (3,3): 71 (3,4): 233 (3,5): 517 (3,6): 676 (3,7): 804 (3,8): 777 (3,9): 617 (3,10): 336 (3,11): 107 (3,12): 78 (3,13): 32 (3,14): 3

witness diameter X 3.1623 · Z 5.0 (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

X checkZ checkqubit site (20)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 89 nearest-neighbor SWAPs per round in total, at most 6 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 code-calculus v0.2rc greedy gated

model Beam 1.0 (claimed, not verified)

date 2026-11-13

notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent and restricted to the qubits the child kept.

family other (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

[[39,4,7]] code-calculus v0.2rc greedy gated

Code-calculus v0.2rc greedy gated campaign (cc-distance-raise). Witness-backed upper bound on distance.

Parity checks

X-checks 17 (max weight 8) · Z-checks 18 (max weight 8)

H_X (17 checks, sparse supports)

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

H_Z (18 checks, sparse supports)

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

Code ID 39-4-7 · download JSON · raw on GitHub