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[[36,28,2]] d ≤
n
36
k
28
d
2
kd²/n
3.11111
w
17
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 10, w_Z = 17 (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)
[0, 2]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[1, 28]
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 4 · H_Z acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7–10 (mean 9.4) · H_Z 10–17 (mean 12.333)
qubit degrees H_X 1–4 (mean 1.306) · H_Z 1–2 (mean 1.028)
trapping sets H_X (1,1)×28 (2,0)×70 (3,0)×177 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 28 (1,2): 6 (1,3): 1 (1,4): 1 (2,0): 70 (2,1): 65 (2,2): 29 (2,3): 23 (3,0): 177 (3,1): 351 (3,2): 396 (3,3): 136 (3,4): 59
trapping sets H_Z (1,1)×35 (2,0)×208 (3,0)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 35 (1,2): 1 (2,0): 208 (2,1): 18 (3,0): 81 (3,1): 848 (3,2): 72

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction C-ADD-CHECK strict improvement (code calculus v0.2rc): added 2 checks killing every minimum-weight logical of parent codes/36-30-1.json
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 > 8 (computed)

How this code was found

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

[[36,28,2]] — C-ADD-CHECK strict improvement of 36-30-1

Direction & hypothesis

Convert the parent (codes/36-30-1.json, d=2) by C-ADD-CHECK strict improvement (Proposition 18.3, code calculus v0.2rc): every minimum-weight logical class of the parent is killed by the added checks, so the child's distance is at least d+1.

What was searched

For d=1 parents: the minimum-weight classes are the bare (weight-1) logicals, enumerated in O(n); the added check supports are their union (plus a GF(2)-solved parity correction when the bare set alone is inadmissible). For d=2 parents: the weight-2 logical classes are enumerated (O(n^2)) and a greedy set-cover over admissible checks kills them within the check budget.

Evidence trail

  • Parent: codes/36-30-1.json.
  • Child witnesses: inherited from the parent where still valid, plus

    • fresh witnesses at the child's distance found by bounded scan.

  • Verified by verify/validate_candidate.py (passed: true), including the

    • random-seed refutation pass.

Dead ends

Parents whose bare-logical set gave an inadmissible check support were fixed by a parity correction solved over GF(2); parents whose system is inconsistent have no single-check fix and were skipped. Multi-check greedy at d=2 succeeds on only a minority of parents (the weight-2 class count grows quickly with n).

Tools

Model: Beam 1.0. Author: @MathysRennela. Construction calculus v0.2rc.

Reproduction

Start from codes/36-30-1.json; add the recorded checks to obtain this entry.

Parity checks

X-checks 5 (max weight 10) · Z-checks 3 (max weight 17)
H_X (5 checks, sparse supports)
[0, 1, 13, 15, 18, 20, 21, 22, 25, 28] [0, 1, 14, 16, 17, 19, 23, 24, 25, 28] [0, 2, 3, 6, 21, 23, 25, 26, 30, 31] [4, 6, 7, 9, 25, 27, 29, 30, 33, 34] [5, 8, 10, 11, 12, 32, 35]
H_Z (3 checks, sparse supports)
[0, 2, 10, 12, 13, 14, 18, 19, 21, 23] [4, 5, 7, 8, 10, 11, 29, 32, 33, 35] [1, 3, 6, 9, 15, 16, 17, 20, 22, 24, 25, 26, 27, 28, 30, 31, 34]
Code ID 36-28-2 · download JSON · raw on GitHub