[[70,24,5]] · QEC Challenge

[[70,24,5]] d =

kd²/n

8.57143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 12, w_Z = 12 (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)

[6, 13, 20, 27, 34]

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)

[0, 14, 23, 30, 37]

certificate exact, d = 5 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

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 12 · H_Z 12

qubit degrees H_X 1–6 (mean 3.771) · H_Z 1–6 (mean 4.114)

trapping sets H_X (1,1)×3 (2,1)×2 (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): 4 (1,3): 21 (1,4): 21 (1,5): 20 (1,6): 1 (2,1): 2 (2,2): 14 (2,3): 38 (2,4): 163 (2,5): 320 (2,6): 353 (2,7): 213 (2,8): 110 (2,9): 10 (3,1): 8 (3,2): 41 (3,3): 292 (3,4): 1038 (3,5): 2502 (3,6): 4442 (3,7): 6016 (3,8): 5769 (3,9): 3950 (3,10): 1859 (3,11): 646 (3,12): 139 (3,13): 33 (3,14): 4

trapping sets H_Z (1,1)×1 (2,2)×9 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 3 (1,3): 15 (1,4): 22 (1,5): 26 (1,6): 3 (2,2): 9 (2,3): 21 (2,4): 83 (2,5): 261 (2,6): 389 (2,7): 314 (2,8): 194 (2,9): 42 (2,10): 2 (3,1): 1 (3,2): 13 (3,3): 116 (3,4): 573 (3,5): 1557 (3,6): 3309 (3,7): 5462 (3,8): 6874 (3,9): 6048 (3,10): 3700 (3,11): 1670 (3,12): 468 (3,13): 100 (3,14): 25 (3,15): 1

Circuit tier

syndrome-extraction memory circuits committed under circuits/70-24-5/ · 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)

[32833, 32837, 33373, 33402, 33404]

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

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

[33226, 33227, 33246, 33259, 33264]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus C-DELETE-CHECK transform of codes/70-22-5.json

model Beam 1.0 (claimed, not verified)

date 2026-11-01

family other (a tag, not a ranking)

locality unrestricted (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

[[70,24,5]] — priced calculus C-DELETE-CHECK transform of 70-22-5.json

Direction & hypothesis

Weight-12 unrestricted CSS cell. Transform a connected CSS parent (70-22-5.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/70-22-5.json. Rule used for this survivor: C-DELETE-CHECK. 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/70-22-5.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/70-22-5.json by applying the C-DELETE-CHECK move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 22 (max weight 12) · Z-checks 24 (max weight 12)

H_X (22 checks, sparse supports)

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

H_Z (24 checks, sparse supports)

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

Code ID 70-24-5 · download JSON · raw on GitHub