[[32,28,1]] · QEC Challenge

[[32,28,1]] d ≤

kd²/n

0.875

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 1, d_Z ≤ 1 · w_X = 19, w_Z = 16 (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)

[20]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[22]

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

check weights H_X 19 · H_Z 14–16 (mean 15.0)

qubit degrees H_X 0–1 (mean 0.594) · H_Z 0–3 (mean 1.406)

trapping sets H_X (1,0)×13 (2,0)×171 (3,1)×969 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 13 (1,1): 19 (2,0): 171 (3,1): 969

trapping sets H_Z (1,0)×1 (2,0)×67 (3,0)×534 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 1 (1,1): 18 (1,2): 12 (1,3): 1 (2,0): 67 (2,1): 153 (2,2): 63 (3,0): 534 (3,1): 1015 (3,2): 663 (3,3): 322

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction code-calculus v0.2rc greedy gated

model Beam 1.0 (claimed, not verified)

builds on arXiv:2504.08887

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

[[32,28,1]] — code-calculus v0.2rc greedy gated

Direction & hypothesis

Frontier gaps in the CSS track via greedy multi-step composition of C-DELETE-CHECK, E-PEEL, and E-PUNCTURE (code-calculus v0.2rc).

What was searched

CSS parents from codes/ with n <= 10, up to 3 greedy steps, budget 16 RIS trials per candidate, sector swap enabled, domination pruning. See the greedy gated campaign (code-calculus v0.2rc).

Evidence trail

Witness-backed upper bound validated by verify/validate_candidate.py (passed: true). Model: Beam 1.0.

Dead ends

Sector-swap-only, strict C-ADD-CHECK addition, and E-GLUE produced no board-advancing codes in this campaign.

Tools

code-calculus-dev (latest), research/kit coordination.

Reproduction

Run the greedy gated campaign (code-calculus v0.2rc) at n<=10, 3 steps, budget 16.

Parity checks

X-checks 1 (max weight 19) · Z-checks 3 (max weight 16)

H_X (1 checks, sparse supports)

[1, 3, 5, 6, 8, 9, 11, 12, 13, 15, 17, 20, 23, 24, 25, 26, 29, 30, 31]

H_Z (3 checks, sparse supports)

[0, 1, 2, 6, 12, 15, 17, 18, 19, 21, 27, 29, 30, 31] [2, 5, 6, 8, 10, 11, 15, 16, 22, 23, 24, 26, 27, 28, 29, 30] [1, 3, 4, 5, 7, 9, 11, 13, 14, 16, 19, 22, 24, 25, 27]

Code ID 32-28-1 · download JSON · raw on GitHub