[[83,55,2]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 24, w_Z = 24 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 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)

[5, 7]

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)

[0, 2]

certificate exact, d = 2 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 2 exists; Z: no logical < 2 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 24 · H_Z 24

qubit degrees H_X 1–9 (mean 4.048) · H_Z 1–9 (mean 4.048)

trapping sets H_X (1,1)×12 (2,0)×4 (3,0)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 12 (1,2): 8 (1,3): 16 (1,4): 8 (1,5): 17 (1,6): 14 (1,7): 6 (1,8): 1 (1,9): 1 (2,0): 4 (2,1): 32 (2,2): 72 (2,3): 140 (2,4): 287 (2,5): 415 (2,6): 486 (2,7): 405 (2,8): 257 (2,9): 118 (2,10): 41 (2,11): 5 (2,12): 1 (3,0): 31 (3,1): 198 (3,2): 829 (3,3): 2406 (3,4): 5677 (3,5): 9719 (3,6): 12855 (3,7): 12749 (3,8): 9549 (3,9): 5551 (3,10): 2346 (3,11): 700 (3,12): 170 (3,13): 22 (3,14): 1

trapping sets H_Z (1,1)×12 (2,0)×4 (3,0)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 12 (1,2): 8 (1,3): 16 (1,4): 8 (1,5): 17 (1,6): 14 (1,7): 6 (1,8): 1 (1,9): 1 (2,0): 4 (2,1): 32 (2,2): 72 (2,3): 140 (2,4): 287 (2,5): 415 (2,6): 486 (2,7): 405 (2,8): 257 (2,9): 118 (2,10): 41 (2,11): 5 (2,12): 1 (3,0): 31 (3,1): 198 (3,2): 829 (3,3): 2406 (3,4): 5677 (3,5): 9719 (3,6): 12855 (3,7): 12749 (3,8): 9549 (3,9): 5551 (3,10): 2346 (3,11): 700 (3,12): 170 (3,13): 22 (3,14): 1

witness diameter X 0.0 · Z 1.0 (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/83-55-2/ · canonical noise recipe, 2 rounds, stim 1.16.0

d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[0, 213]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[3531, 3543]

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/85-37-5.json: the parent's coordinates with the entries of the removed qubits (parent indices 11, 16) dropped, since this puncture chain keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04

check (X = Z, self-dual)qubit site (44)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 58 nearest-neighbor SWAPs per round in total, at most 4 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/85-37-5.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

[[83,55,2]] — priced calculus E-PUNCTURE transform of 85-37-5.json

Direction & hypothesis

Weight-24 unrestricted CSS cell. Transform a connected CSS parent (85-37-5.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/85-37-5.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/85-37-5.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/85-37-5.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 14 (max weight 24) · Z-checks 14 (max weight 24)

H_X (14 checks, sparse supports)

[0, 2, 3, 9, 11, 12, 16, 19, 20, 25, 28, 29, 30, 32, 34, 36, 41, 44, 46, 47, 50, 52, 54, 59] [1, 3, 4, 10, 12, 13, 17, 20, 21, 26, 29, 30, 31, 33, 35, 37, 42, 45, 47, 48, 51, 53, 55, 60] [5, 7, 8, 13, 15, 16, 21, 24, 25, 30, 33, 34, 35, 37, 39, 41, 46, 49, 51, 52, 55, 57, 59, 64] [6, 8, 9, 14, 16, 17, 22, 25, 26, 31, 34, 35, 36, 38, 40, 42, 47, 50, 52, 53, 56, 58, 60, 65] [10, 11, 12, 17, 20, 21, 26, 29, 30, 35, 38, 39, 40, 42, 44, 46, 51, 54, 56, 57, 60, 62, 64, 69] [11, 13, 14, 19, 22, 23, 28, 31, 32, 37, 40, 41, 42, 44, 46, 48, 53, 56, 58, 59, 62, 64, 66, 71] [14, 15, 16, 22, 25, 26, 31, 34, 35, 40, 43, 44, 45, 47, 49, 51, 56, 59, 61, 62, 65, 67, 69, 74] [15, 17, 18, 24, 27, 28, 33, 36, 37, 42, 45, 46, 47, 49, 51, 53, 58, 61, 63, 64, 67, 69, 71, 76] [16, 18, 19, 25, 28, 29, 34, 37, 38, 43, 46, 47, 48, 50, 52, 54, 59, 62, 64, 65, 68, 70, 72, 77] [17, 19, 20, 26, 29, 30, 35, 38, 39, 44, 47, 48, 49, 51, 53, 55, 60, 63, 65, 66, 69, 71, 73, 78] [18, 20, 21, 27, 30, 31, 36, 39, 40, 45, 48, 49, 50, 52, 54, 56, 61, 64, 66, 67, 70, 72, 74, 79] [19, 21, 22, 28, 31, 32, 37, 40, 41, 46, 49, 50, 51, 53, 55, 57, 62, 65, 67, 68, 71, 73, 75, 80] [20, 22, 23, 29, 32, 33, 38, 41, 42, 47, 50, 51, 52, 54, 56, 58, 63, 66, 68, 69, 72, 74, 76, 81] [21, 23, 24, 30, 33, 34, 39, 42, 43, 48, 51, 52, 53, 55, 57, 59, 64, 67, 69, 70, 73, 75, 77, 82]

H_Z (14 checks, sparse supports)

[0, 2, 3, 9, 11, 12, 16, 19, 20, 25, 28, 29, 30, 32, 34, 36, 41, 44, 46, 47, 50, 52, 54, 59] [1, 3, 4, 10, 12, 13, 17, 20, 21, 26, 29, 30, 31, 33, 35, 37, 42, 45, 47, 48, 51, 53, 55, 60] [5, 7, 8, 13, 15, 16, 21, 24, 25, 30, 33, 34, 35, 37, 39, 41, 46, 49, 51, 52, 55, 57, 59, 64] [6, 8, 9, 14, 16, 17, 22, 25, 26, 31, 34, 35, 36, 38, 40, 42, 47, 50, 52, 53, 56, 58, 60, 65] [10, 11, 12, 17, 20, 21, 26, 29, 30, 35, 38, 39, 40, 42, 44, 46, 51, 54, 56, 57, 60, 62, 64, 69] [11, 13, 14, 19, 22, 23, 28, 31, 32, 37, 40, 41, 42, 44, 46, 48, 53, 56, 58, 59, 62, 64, 66, 71] [14, 15, 16, 22, 25, 26, 31, 34, 35, 40, 43, 44, 45, 47, 49, 51, 56, 59, 61, 62, 65, 67, 69, 74] [15, 17, 18, 24, 27, 28, 33, 36, 37, 42, 45, 46, 47, 49, 51, 53, 58, 61, 63, 64, 67, 69, 71, 76] [16, 18, 19, 25, 28, 29, 34, 37, 38, 43, 46, 47, 48, 50, 52, 54, 59, 62, 64, 65, 68, 70, 72, 77] [17, 19, 20, 26, 29, 30, 35, 38, 39, 44, 47, 48, 49, 51, 53, 55, 60, 63, 65, 66, 69, 71, 73, 78] [18, 20, 21, 27, 30, 31, 36, 39, 40, 45, 48, 49, 50, 52, 54, 56, 61, 64, 66, 67, 70, 72, 74, 79] [19, 21, 22, 28, 31, 32, 37, 40, 41, 46, 49, 50, 51, 53, 55, 57, 62, 65, 67, 68, 71, 73, 75, 80] [20, 22, 23, 29, 32, 33, 38, 41, 42, 47, 50, 51, 52, 54, 56, 58, 63, 66, 68, 69, 72, 74, 76, 81] [21, 23, 24, 30, 33, 34, 39, 42, 43, 48, 51, 52, 53, 55, 57, 59, 64, 67, 69, 70, 73, 75, 77, 82]

Code ID 83-55-2 · download JSON · raw on GitHub