[[42,8,3]] · QEC Challenge

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[[42,8,3]] d =

kd²/n

1.714

X/Z

0.0011

6.3246

layers

swaps

106

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[4, 11, 18]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 3, 15]

certificate exact, d = 3 · scipy/HiGHS MILP

X: no logical < 3 exists; Z: no logical < 3 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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×21 (2,2)×21 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 21 (1,4): 21 (2,2): 21 (2,4): 231 (3,0): 28 (3,4): 1260 (3,6): 672

trapping sets H_Z (1,2)×21 (2,2)×21 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 21 (1,4): 21 (2,2): 21 (2,4): 231 (3,0): 28 (3,4): 1260 (3,6): 672

witness diameter X 5.099 · 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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/42-8-3/ · canonical noise recipe, 3 rounds, stim 1.16.0

d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[4347, 4348, 4377]

d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[0, 6, 156]

ler/round (X) 0.00466 95% CI [0.00414, 0.00524] · 277/2×10⁴ shots · decoder bposd-cs-10 at p=0.001

ler/round (Z) 0.00429 95% CI [0.00379, 0.00485] · 255/2×10⁴ shots · decoder bposd-cs-10 at p=0.001

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 (21)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 106 nearest-neighbor SWAPs per round in total, at most 5 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 Wang, Renren and Pryadko, Leonid P.

provenance literature baseline

construction Generalized bicycle code, circulant size 21, a(x) = 1 + x³ + x⁶ + x¹², b(x) = 1 + x⁷.

model classical construction (no AI model)

reference arXiv:2203.17216

date 2022-03-31

notes Distance d=3 from arXiv:2203.17216; witness here is a decoder-found logical operator of that weight (upper bound). Layout certification: a verifier-accepted honest bilayer layout (stacked 3x7 grid, 2 qubits per site, layers=2, measured interaction radius ~6.325) was found by @mathysrennela (Aug 2026). It earns the code the local-2d-bilayer class and makes the site's geometric efficiency g = 4kd^2/(n rho^2 r^4) computable. Code authorship unchanged; layout credit to @mathysrennela.

family generalized 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 21 (max weight 6) · Z-checks 21 (max weight 6)

H_X (21 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 42-8-3 · download JSON · raw on GitHub