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[[62,31,3]] d =
n
62
k
31
d
3
kd²/n
4.5
w
16
X/Z
1
g
0.0039
r
5.831
layers
2
swaps
119

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 16, w_Z = 16 (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 found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01
witness operator (support, 3 qubits)
[14, 45, 51]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01
witness operator (support, 3 qubits)
[10, 41, 49]
certificate exact, d = 3 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI
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 16 · H_Z 15–16 (mean 15.667)
qubit degrees H_X 2–8 (mean 4.129) · H_Z 1–7 (mean 3.79)
trapping sets H_X (1,2)×6 (2,1)×6 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 6 (1,3): 15 (1,4): 19 (1,5): 14 (1,6): 4 (1,7): 3 (1,8): 1 (2,1): 6 (2,2): 23 (2,3): 91 (2,4): 179 (2,5): 282 (2,6): 290 (2,7): 218 (2,8): 123 (2,9): 55 (2,10): 21 (2,11): 1 (2,12): 1 (3,0): 2 (3,1): 29 (3,2): 167 (3,3): 697 (3,4): 1902 (3,5): 3615 (3,6): 5337 (3,7): 5769 (3,8): 4769 (3,9): 3061 (3,10): 1481 (3,11): 538 (3,12): 137 (3,13): 21 (3,14): 5
trapping sets H_Z (1,1)×5 (2,1)×8 (3,0)×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): 5 (1,2): 4 (1,3): 16 (1,4): 18 (1,5): 13 (1,6): 5 (1,7): 1 (2,1): 8 (2,2): 27 (2,3): 115 (2,4): 213 (2,5): 282 (2,6): 263 (2,7): 179 (2,8): 80 (2,9): 33 (2,10): 2 (3,0): 4 (3,1): 42 (3,2): 272 (3,3): 967 (3,4): 2397 (3,5): 4099 (3,6): 5265 (3,7): 5107 (3,8): 3773 (3,9): 1947 (3,10): 759 (3,11): 212 (3,12): 41 (3,13): 6
witness diameter X 4.1231 · Z 4.2426 (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/62-31-3/ · 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)
[487, 499, 577]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[16219, 16224, 16333]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

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 @vprusso · Inherited from the laid-out parent codes/64-20-8.json: the parent's coordinates with the entries of the removed qubits (parent indices 2, 61) dropped, since this peel chain keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 5.831
X checkZ checkqubit site (32)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 119 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 priced calculus E-PEEL transform of codes/64-20-8.json
model Beam 1.0 (claimed, not verified)
date 2026-11-01
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

[[62,31,3]] — priced calculus E-PEEL transform of 64-20-8.json

Direction & hypothesis

Weight-16 unrestricted CSS cell. Transform a connected CSS parent (64-20-8.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Single-step and depth-2 deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev. Parent: codes/64-20-8.json. Rule used for this survivor: E-PEEL. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side (research/kit/surrogate.lightest_logical, seeds 0/1), then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. The gate's random-seed refutation found no lighter logical. Parent entry: codes/64-20-8.json.

Dead ends

Other single-step moves from the same parents were discarded as disconnected, bare-qubit children, dominated candidates, or refuted by the gate.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer. Approx compute: CPU only.

Reproduction

Rebuild H_X/H_Z from codes/64-20-8.json by applying the E-PEEL move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 16 (max weight 16) · Z-checks 15 (max weight 16)
H_X (16 checks, sparse supports)
[3, 5, 12, 14, 20, 22, 27, 29, 31, 33, 40, 42, 48, 50, 55, 57] [4, 6, 11, 13, 15, 17, 24, 26, 31, 33, 40, 42, 52, 54, 59, 60] [32, 34, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 56, 58, 59, 60] [0, 1, 7, 8, 17, 18, 25, 26, 35, 36, 43, 44, 53, 54, 60, 61] [2, 4, 9, 11, 18, 20, 25, 27, 31, 37, 40, 46, 47, 53, 56, 61] [1, 6, 9, 11, 18, 20, 23, 29, 34, 36, 39, 45, 48, 54, 57, 59] [3, 5, 11, 13, 16, 18, 24, 26, 32, 34, 40, 42, 51, 53, 59, 60] [0, 2, 11, 14, 15, 18, 27, 30, 35, 38, 39, 42, 51, 54, 55, 58] [16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46] [2, 3, 8, 13, 16, 21, 26, 27, 31, 38, 41, 44, 49, 52, 55, 61] [3, 5, 7, 9, 16, 18, 28, 30, 31, 33, 43, 45, 52, 54, 56, 58] [8, 9, 11, 14, 16, 17, 19, 22, 39, 42, 44, 45, 47, 50, 52, 53] [7, 8, 9, 10, 19, 20, 21, 22, 43, 44, 45, 46, 47, 48, 49, 50] [1, 2, 4, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 28, 30] [1, 2, 8, 10, 15, 17, 23, 25, 32, 34, 40, 42, 47, 49, 55, 57] [3, 5, 12, 14, 20, 22, 27, 29, 32, 34, 39, 41, 47, 49, 56, 58]
H_Z (15 checks, sparse supports)
[1, 5, 9, 12, 18, 19, 23, 30, 31, 38, 42, 43, 49, 52, 56, 60] [4, 6, 8, 10, 20, 22, 24, 26, 32, 34, 44, 46, 48, 50, 61] [3, 4, 5, 6, 19, 20, 21, 22, 39, 40, 41, 42, 55, 56, 57, 58] [8, 9, 12, 13, 16, 17, 20, 21, 32, 33, 36, 37, 56, 57, 60] [7, 8, 13, 14, 25, 26, 27, 28, 31, 32, 37, 38, 49, 50, 51, 52] [1, 2, 12, 14, 15, 17, 27, 29, 31, 33, 43, 45, 48, 50, 61] [1, 3, 7, 12, 16, 19, 23, 28, 34, 37, 41, 46, 50, 53, 57, 61] [1, 3, 10, 13, 15, 20, 25, 30, 34, 37, 40, 43, 49, 54, 55] [0, 2, 4, 5, 24, 25, 27, 30, 32, 33, 35, 38, 55, 58, 60] [1, 6, 8, 14, 16, 22, 24, 30, 33, 35, 41, 43, 49, 51, 57, 59] [9, 10, 11, 12, 23, 24, 29, 30, 41, 42, 43, 44, 55, 56, 60, 61] [0, 4, 10, 13, 15, 20, 26, 29, 33, 38, 40, 43, 49, 54, 56, 59] [3, 6, 7, 10, 16, 17, 28, 29, 36, 37, 40, 41, 47, 50, 59, 61] [0, 5, 7, 13, 17, 19, 25, 27, 34, 36, 42, 44, 48, 54, 56, 61] [4, 6, 8, 10, 19, 21, 23, 25, 35, 37, 39, 41, 52, 54, 56, 58]
Code ID 62-31-3 · download JSON · raw on GitHub