[[71,39,4]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (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, 4 qubits)

[2, 5, 27, 68]

d_Z 4 · witness weight 4 (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, 4 qubits)

[2, 8, 51, 58]

certificate exact, d = 4 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 4 exists; Z: no logical < 4 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 1–10 (mean 6.31) · H_Z 1–10 (mean 6.31)

trapping sets H_X (1,1)×3 (2,1)×4 (3,1)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 3 (1,2): 5 (1,3): 6 (1,4): 7 (1,5): 7 (1,6): 5 (1,7): 6 (1,8): 10 (1,9): 18 (1,10): 4 (2,1): 4 (2,2): 13 (2,3): 33 (2,4): 80 (2,5): 152 (2,6): 250 (2,7): 362 (2,8): 438 (2,9): 390 (2,10): 242 (2,11): 106 (2,12): 37 (2,13): 10 (3,1): 8 (3,2): 83 (3,3): 436 (3,4): 1477 (3,5): 3473 (3,6): 6464 (3,7): 9213 (3,8): 10313 (3,9): 8900 (3,10): 6068 (3,11): 3239 (3,12): 1296 (3,13): 317 (3,14): 66 (3,15): 8

trapping sets H_Z (1,1)×3 (2,1)×4 (3,1)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 3 (1,2): 5 (1,3): 6 (1,4): 7 (1,5): 7 (1,6): 5 (1,7): 6 (1,8): 10 (1,9): 18 (1,10): 4 (2,1): 4 (2,2): 13 (2,3): 33 (2,4): 80 (2,5): 152 (2,6): 250 (2,7): 362 (2,8): 438 (2,9): 390 (2,10): 242 (2,11): 106 (2,12): 37 (2,13): 10 (3,1): 8 (3,2): 83 (3,3): 436 (3,4): 1477 (3,5): 3473 (3,6): 6464 (3,7): 9213 (3,8): 10313 (3,9): 8900 (3,10): 6068 (3,11): 3239 (3,12): 1296 (3,13): 317 (3,14): 66 (3,15): 8

witness diameter X 3.0 · Z 7.0711 (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/71-39-4/ · canonical noise recipe, 4 rounds, stim 1.16.0

d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[15322, 15344, 16538, 16869]

d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[40923, 40932, 40939, 40950]

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 53, 54) 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 16 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

[[71,39,4]] — 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 16 (max weight 28) · Z-checks 16 (max weight 28)

H_X (16 checks, sparse supports)

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

H_Z (16 checks, sparse supports)

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

Code ID 71-39-4 · download JSON · raw on GitHub