[[64,23,5]] · QEC Challenge

[[64,23,5]] d ≤

kd²/n

8.98438

X/Z

1.2

Share this result

Distance

X/Z asymmetry 1.2 · d_X ≤ 6, d_Z ≤ 5 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (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, 6 qubits)

[13, 31, 36, 41, 54, 59]

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, 9, 31, 32, 38]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 10 · H_Z 10

qubit degrees H_X 1–5 (mean 3.125) · H_Z 2–5 (mean 3.281)

trapping sets H_X (1,1)×1 (2,1)×2 (3,1)×20 (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): 15 (1,3): 26 (1,4): 19 (1,5): 3 (2,1): 2 (2,2): 28 (2,3): 137 (2,4): 246 (2,5): 254 (2,6): 125 (2,7): 29 (2,8): 1 (3,1): 20 (3,2): 157 (3,3): 663 (3,4): 1839 (3,5): 3382 (3,6): 3665 (3,7): 2588 (3,8): 1248 (3,9): 421 (3,10): 93 (3,11): 10 (3,12): 3

trapping sets H_Z (1,2)×13 (2,2)×18 (3,1)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 13 (1,3): 24 (1,4): 23 (1,5): 4 (2,2): 18 (2,3): 105 (2,4): 238 (2,5): 269 (2,6): 179 (2,7): 46 (2,8): 2 (3,1): 10 (3,2): 103 (3,3): 464 (3,4): 1480 (3,5): 3084 (3,6): 4010 (3,7): 3309 (3,8): 1776 (3,9): 699 (3,10): 168 (3,11): 24 (3,12): 5

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus C-DELETE-CHECK transform of codes/64-22-6.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

[[64,23,5]] — priced calculus C-DELETE-CHECK transform of 64-22-6.json

Direction & hypothesis

Weight-10 unrestricted CSS cell. Transform a connected CSS parent (64-22-6.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/64-22-6.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/64-22-6.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/64-22-6.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 20 (max weight 10) · Z-checks 21 (max weight 10)

H_X (20 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 64-23-5 · download JSON · raw on GitHub