[[71,55,2]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 28, w_Z = 28 (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)

[54, 66]

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)

[46, 49]

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 28 · H_Z 28

qubit degrees H_X 1–7 (mean 3.155) · H_Z 1–7 (mean 3.155)

trapping sets H_X (1,1)×12 (2,0)×17 (3,0)×196 (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): 19 (1,3): 11 (1,4): 11 (1,5): 12 (1,6): 5 (1,7): 1 (2,0): 17 (2,1): 90 (2,2): 221 (2,3): 449 (2,4): 486 (2,5): 344 (2,6): 133 (2,7): 18 (3,0): 196 (3,1): 1611 (3,2): 5445 (3,3): 10147 (3,4): 11523 (3,5): 8476 (3,6): 3777 (3,7): 966 (3,8): 134

trapping sets H_Z (1,1)×12 (2,0)×17 (3,0)×196 (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): 19 (1,3): 11 (1,4): 11 (1,5): 12 (1,6): 5 (1,7): 1 (2,0): 17 (2,1): 90 (2,2): 221 (2,3): 449 (2,4): 486 (2,5): 344 (2,6): 133 (2,7): 18 (3,0): 196 (3,1): 1611 (3,2): 5445 (3,3): 10147 (3,4): 11523 (3,5): 8476 (3,6): 3777 (3,7): 966 (3,8): 134

witness diameter X 2.2361 · Z 2.2361 (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/71-55-2/ · 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)

[107, 108]

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

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

[1588, 1589]

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/73-37-6.json: the parent's coordinates with the entries of the removed qubits (parent indices 15, 42) 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 (38)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 10 nearest-neighbor SWAPs per round in total, at most 3 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/73-37-6.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

[[71,55,2]] — priced calculus E-PUNCTURE transform of 73-37-6.json

Direction & hypothesis

Weight-28 unrestricted CSS cell. Transform a connected CSS parent (73-37-6.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/73-37-6.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/73-37-6.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/73-37-6.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 8 (max weight 28) · Z-checks 8 (max weight 28)

H_X (8 checks, sparse supports)

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

H_Z (8 checks, sparse supports)

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

Code ID 71-55-2 · download JSON · raw on GitHub