[[30,10,3]] · QEC Challenge

[[30,10,3]] d =

kd²/n

3.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[1, 5, 14]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[7, 13, 27]

certificate exact, d = 3 · 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 < 3 exists; Z: no logical < 3 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 1–2 (mean 1.667) · H_Z 1–2 (mean 1.667)

trapping sets H_X (1,1)×10 (2,1)×40 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 20 (2,1): 40 (2,2): 60 (3,0): 30 (3,1): 120 (3,2): 150 (3,3): 60 (3,4): 40

trapping sets H_Z (1,1)×10 (2,1)×40 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 20 (2,1): 40 (2,2): 60 (3,0): 30 (3,1): 120 (3,2): 150 (3,3): 60 (3,4): 40

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK b9 from 30-9-3.json

model Beam 1.0 (claimed, not verified)

date 2026-10-25

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[30,10,3]] — priced-calculus C-DELETE-CHECK

Direction & hypothesis

Aimed at a thin (n,k,d) cell. Derived by a single C-DELETE-CHECK move of the priced construction calculus from a parent entry, keeping only instances with a connected Tanner graph and fresh X/Z witnesses.

What was searched

Single-move C-DELETE-CHECK sweep over challenge parents with n <= 40, d >= 3 (2-hour campaign, 2026-10-25); survivors screened for X and Z logical witnesses up to weight 6 and validated with verify/validate_candidate.py.

Evidence trail

Witness-backed upper bounds: d_X <= 3, d_Z <= 3, so d <= 3.

Dead ends

Disconnected or witness-less instances were discarded.

Tools

Model: Beam 1.0. Priced construction calculus consumer (C-DELETE-CHECK cell). Challenge verifier verify/qldpc_verify.py.

Reproduction

Operation: C-DELETE-CHECK b9 from 30-9-3.json. Witnesses as stored in the entry.

Parity checks

X-checks 10 (max weight 5) · Z-checks 10 (max weight 5)

H_X (10 checks, sparse supports)

[2, 3, 5, 8, 26] [6, 7, 11, 13, 18] [4, 5, 16, 17, 20] [2, 19, 20, 22, 25] [15, 19, 21, 26, 29] [1, 6, 9, 16, 24] [10, 12, 13, 27, 29] [7, 14, 21, 25, 27] [1, 10, 18, 23, 28] [0, 3, 9, 17, 23]

H_Z (10 checks, sparse supports)

[2, 3, 9, 22, 24] [11, 13, 19, 22, 29] [1, 10, 14, 24, 27] [0, 15, 17, 19, 20] [4, 5, 14, 21, 26] [0, 6, 9, 12, 13] [6, 7, 16, 20, 25] [1, 5, 8, 16, 28] [3, 10, 23, 26, 29] [7, 15, 18, 21, 28]

Code ID 30-10-3-b · download JSON · raw on GitHub