[[69,50,1]] · QEC Challenge

[[69,50,1]] d ≤

kd²/n

0.724638

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 1, d_Z ≤ 1 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[4]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[27]

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 6 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 8 · H_Z 5–8 (mean 7.636)

qubit degrees H_X 0–3 (mean 0.928) · H_Z 0–2 (mean 1.217)

trapping sets H_X (1,0)×19 (2,0)×71 (3,0)×225 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 19 (1,1): 37 (1,2): 12 (1,3): 1 (2,0): 71 (2,1): 103 (2,2): 43 (2,3): 7 (3,0): 225 (3,1): 386 (3,2): 338 (3,3): 170 (3,4): 45 (3,5): 7

trapping sets H_Z (1,0)×8 (2,0)×53 (3,0)×278 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 8 (1,1): 38 (1,2): 23 (2,0): 53 (2,1): 156 (2,2): 70 (3,0): 278 (3,1): 524 (3,2): 407 (3,3): 226 (3,4): 53

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction code-calculus v0.2rc E-PUNCTURE

model Beam 1.0 (claimed, not verified)

builds on codes/72-38-3.json

date 2026-11-12

notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger.

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

[[69,50,1]] greedy (target from 72-39-1

Direction & hypothesis

Unrestricted-weight CSS parents with 61 <= n <= 100, searched with the priced construction calculus (code-calculus v0.2rc) greedy multi-step composition. The rule used on this step is E-PUNCTURE.

What was searched

Greedy multi-step composition of C-DELETE-CHECK / E-PEEL / E-PUNCTURE with sector-targeted planning, sector swap, bounded support search, and Pareto ranking. This candidate is step (target of a chain from codes/72-39-1.json (target sector X).

Evidence trail

Staged deep-verify (2k -> 20k -> 200k -> 2M trials, multi-seed) kept the screened distance unchanged; the witness-backed upper bound is d_X <= 1, d_Z <= 1. No downgrade at any depth.

Dead ends

Parents whose display carries a dependent generator produce no derivable child; candidates whose deep-verified slug collides with an existing board entry are skipped.

Tools

Model: Beam 1.0. Harness: code-calculus v0.2rc greedy gated campaign, with the verifier's staged deep-verify used before submission.

Reproduction

Start from codes/72-39-1.json, apply E-PUNCTURE on the target sector, and repeat greedily with Pareto ranking and staged deep-verify.

Parity checks

X-checks 8 (max weight 8) · Z-checks 11 (max weight 8)

H_X (8 checks, sparse supports)

[21, 29, 37, 38, 39, 54, 59, 68] [6, 7, 14, 15, 45, 64, 65, 67] [37, 41, 48, 49, 50, 63, 64, 66] [3, 45, 46, 54, 55, 57, 58, 60] [8, 18, 20, 24, 33, 36, 40, 49] [12, 18, 25, 35, 46, 50, 67, 68] [4, 13, 18, 19, 38, 42, 52, 53] [1, 2, 5, 23, 32, 34, 36, 60]

H_Z (11 checks, sparse supports)

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

Code ID 69-50-1 · download JSON · raw on GitHub