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[[62,35,2]] d =
n
62
k
35
d
2
kd²/n
2.25806
w
8
X/Z
1.5
g
0.0027
r
5.3852
layers
2
swaps
80

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Distance

X/Z asymmetry 1.5 · d_X = 3, d_Z = 2 · 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 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)
[18, 45, 58]
d_Z 2 · witness weight 2 (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, 2 qubits)
[4, 12]
certificate exact, d = 2 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: graphlike_A and graphlike_B: d_X = 3, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 2); words of weight 3 rechecked; Z: graphlike_A and graphlike_B: d_Z = 2, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 1); words of weight 2 rechecked

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 7–8 (mean 7.929)
qubit degrees H_X 1–2 (mean 1.677) · H_Z 1–2 (mean 1.79)
trapping sets H_X (1,1)×20 (2,0)×7 (3,0)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 20 (1,2): 42 (2,0): 7 (2,1): 126 (2,2): 231 (3,0): 84 (3,1): 714 (3,2): 1302 (3,3): 336 (3,4): 350
trapping sets H_Z (1,1)×13 (2,1)×91 (3,0)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 13 (1,2): 49 (2,1): 91 (2,2): 294 (3,0): 42 (3,1): 546 (3,2): 1764 (3,3): 273 (3,4): 490
witness diameter X 7.0 · Z 2.0 (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
layout contributed by @vprusso · Inherited from the laid-out parent codes/64-34-4.json: the parent's coordinates with the entries of the removed qubits (parent indices 7, 13) 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 = 5.385
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 80 nearest-neighbor SWAPs per round in total, at most 7 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/64-34-4.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

[[62,35,2]] — priced calculus E-PEEL transform of 64-34-4.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (64-34-4.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/64-34-4.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/64-34-4.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/64-34-4.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 13 (max weight 8) · Z-checks 14 (max weight 8)
H_X (13 checks, sparse supports)
[0, 8, 16, 25, 34, 43, 52, 61] [1, 7, 17, 24, 35, 42, 53, 60] [2, 10, 14, 23, 36, 45, 50, 59] [3, 9, 15, 22, 37, 44, 51, 58] [5, 11, 21, 28, 31, 38, 49, 56] [6, 13, 18, 27, 32, 41, 46, 55] [0, 9, 18, 28, 33, 39, 53, 59] [1, 10, 19, 29, 32, 38, 52, 58] [2, 7, 20, 26, 31, 41, 51, 61] [3, 8, 21, 27, 30, 40, 50, 60] [4, 12, 14, 24, 37, 43, 49, 55] [5, 13, 15, 25, 36, 42, 48, 54] [6, 11, 16, 22, 35, 45, 47, 57]
H_Z (14 checks, sparse supports)
[0, 8, 18, 27, 36, 45, 48, 57] [1, 7, 19, 26, 37, 44, 49, 56] [2, 10, 20, 29, 34, 43, 46, 55] [3, 9, 21, 28, 35, 42, 47, 54] [4, 14, 23, 32, 41, 52, 61] [5, 11, 15, 22, 33, 40, 53, 60] [6, 13, 16, 25, 30, 39, 50, 59] [0, 9, 17, 23, 37, 43, 50, 60] [1, 10, 16, 22, 36, 42, 51, 61] [2, 7, 15, 25, 35, 45, 52, 58] [3, 8, 14, 24, 34, 44, 53, 59] [4, 12, 21, 27, 33, 39, 46, 56] [5, 13, 20, 26, 32, 38, 47, 57] [6, 11, 19, 29, 31, 41, 48, 54]
Code ID 62-35-2 · download JSON · raw on GitHub