[[23,12,2]] · QEC Challenge

[[23,12,2]] d ≤

kd²/n

2.08696

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[3, 20]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[10, 12]

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

check weights H_X 5–6 (mean 5.8) · H_Z 5–6 (mean 5.5)

qubit degrees H_X 1–2 (mean 1.261) · H_Z 1–2 (mean 1.435)

trapping sets H_X (1,1)×17 (2,0)×21 (3,0)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 17 (1,2): 6 (2,0): 21 (2,1): 40 (2,2): 9 (3,0): 66 (3,1): 73 (3,2): 60 (3,3): 29 (3,4): 2

trapping sets H_Z (1,1)×13 (2,0)×9 (3,0)×43 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 13 (1,2): 10 (2,0): 9 (2,1): 40 (2,2): 26 (3,0): 43 (3,1): 112 (3,2): 73 (3,3): 46 (3,4): 17

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)

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 ≤ 6 (computed)

How this code was found

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

[[23,12,2]] — 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 5 (max weight 6) · Z-checks 6 (max weight 6)

H_X (5 checks, sparse supports)

[4, 6, 11, 15, 16, 17] [1, 8, 10, 12, 13] [9, 13, 14, 17, 18, 20] [3, 4, 5, 7, 8, 19] [0, 2, 5, 9, 21, 22]

H_Z (6 checks, sparse supports)

[1, 2, 3, 5, 13, 20] [4, 6, 10, 12, 19] [1, 8, 14, 16, 17, 19] [2, 4, 7, 9, 16, 18] [0, 11, 17, 18, 22] [5, 6, 7, 15, 21]

Code ID 23-12-2 · download JSON · raw on GitHub