[[37,18,2]] · QEC Challenge

[[37,18,2]] d ≤

kd²/n

1.94595

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness found by @mathysrennela · gf2_fast · found at 2×10⁶ trials · 2026-11-12

witness operator (support, 2 qubits)

[2, 30]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[3, 34]

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 acyclic (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 1.351) · H_Z 1–2 (mean 1.216)

trapping sets H_X (1,1)×24 (2,0)×24 (3,0)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 24 (1,2): 13 (2,0): 24 (2,1): 48 (2,2): 28 (3,0): 36 (3,1): 128 (3,2): 100 (3,3): 24 (3,4): 20

trapping sets H_Z (1,1)×29 (2,0)×36 (3,0)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 29 (1,2): 8 (2,0): 36 (2,1): 44 (2,2): 10 (3,0): 60 (3,1): 76 (3,2): 60 (3,3): 18 (3,4): 4

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction code-calculus v0.2rc greedy gated

model Beam 1.0 (claimed, not verified)

builds on https://qecdb.org/codes/67a4b0149edf81e4b7e670ae (isomorphic record; research/provenance/derived.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 ≤ 6 (computed)

How this code was found

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

[[37,18,2]] — 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 10 (max weight 5) · Z-checks 9 (max weight 5)

H_X (10 checks, sparse supports)

[14, 23, 29, 31, 36] [4, 8, 9, 20, 21] [4, 10, 17, 26, 28] [5, 11, 27, 30, 33] [10, 16, 22, 24, 29] [0, 1, 2, 6, 25] [8, 16, 19, 25, 30] [1, 3, 15, 22, 34] [7, 13, 18, 19, 31] [12, 24, 27, 32, 35]

H_Z (9 checks, sparse supports)

[4, 7, 8, 19, 26] [8, 12, 16, 20, 24] [13, 17, 23, 28, 31] [4, 10, 14, 21, 29] [11, 22, 24, 27, 34] [2, 25, 27, 30, 35] [0, 1, 12, 15, 32] [9, 18, 20, 31, 36] [1, 3, 5, 6, 33]

Code ID 37-18-2 · download JSON · raw on GitHub