[[65,26,5]] · QEC Challenge

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[[65,26,5]] d =

kd²/n

10.0

X/Z

0.0049

6.7082

layers

swaps

137

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 19, w_Z = 19 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (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, 5 qubits)

[5, 21, 23, 31, 63]

d_Z 5 · witness weight 5 (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, 5 qubits)

[4, 8, 24, 34, 42]

certificate exact, d = 5 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 5 exists; Z: no logical < 5 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 19 · H_Z 19

qubit degrees H_X 1–10 (mean 6.138) · H_Z 1–9 (mean 5.262)

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

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

(1,1): 1 (1,2): 4 (1,3): 5 (1,4): 6 (1,5): 9 (1,6): 9 (1,7): 13 (1,8): 4 (1,9): 11 (1,10): 3 (2,2): 3 (2,3): 9 (2,4): 37 (2,5): 100 (2,6): 153 (2,7): 197 (2,8): 270 (2,9): 277 (2,10): 271 (2,11): 145 (2,12): 82 (2,13): 53 (2,14): 34 (2,15): 13 (2,16): 1 (3,2): 3 (3,3): 50 (3,4): 263 (3,5): 791 (3,6): 1812 (3,7): 3243 (3,8): 4843 (3,9): 5964 (3,10): 6096 (3,11): 5337 (3,12): 4074 (3,13): 2494 (3,14): 1344 (3,15): 506 (3,16): 186 (3,17): 73 (3,18): 20 (3,19): 1 (3,20): 1

trapping sets H_Z (1,1)×2 (2,1)×6 (3,1)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 2 (1,2): 9 (1,3): 4 (1,4): 8 (1,5): 8 (1,6): 12 (1,7): 12 (1,8): 8 (1,9): 2 (2,1): 6 (2,2): 6 (2,3): 22 (2,4): 96 (2,5): 144 (2,6): 257 (2,7): 282 (2,8): 260 (2,9): 206 (2,10): 130 (2,11): 92 (2,12): 21 (2,13): 10 (2,14): 1 (2,15): 2 (3,1): 2 (3,2): 51 (3,3): 240 (3,4): 867 (3,5): 1908 (3,6): 3793 (3,7): 5282 (3,8): 6115 (3,9): 5862 (3,10): 4599 (3,11): 3094 (3,12): 1417 (3,13): 664 (3,14): 182 (3,15): 74 (3,16): 14 (3,17): 2

witness diameter X 5.0 · 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/65-26-5/ · canonical noise recipe, 5 rounds, stim 1.16.0

d_circ ≤ 5 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 5)

[42940, 42947, 42952, 42969, 42985]

d_circ^Z 5 · fault-set witness of 5 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 5)

[16805, 16817, 16985, 17011, 17347]

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/66-20-7.json: the parent's coordinates with the entry of the removed qubit (parent index 29) dropped, since this puncture child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04

X checkZ checkqubit site (33)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 137 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-PUNCTURE transform of codes/66-20-7.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

[[65,26,5]] — priced calculus E-PUNCTURE transform of 66-20-7.json

Direction & hypothesis

Weight-19 unrestricted CSS cell. Transform a connected CSS parent (66-20-7.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/66-20-7.json. Rule used for this survivor: E-PUNCTURE. 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/66-20-7.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/66-20-7.json by applying the E-PUNCTURE move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 21 (max weight 19) · Z-checks 18 (max weight 19)

H_X (21 checks, sparse supports)

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

H_Z (18 checks, sparse supports)

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

Code ID 65-26-5 · download JSON · raw on GitHub