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[[32,26,2]] d ≤
n
32
k
26
d
2
kd²/n
3.25
w
19
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 12]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 2]
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 14–19 (mean 16.5) · H_Z 2–16 (mean 11.75)
qubit degrees H_X 1–2 (mean 1.031) · H_Z 1–3 (mean 1.469)
trapping sets H_X (1,1)×31 (2,0)×231 (3,0)×234 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 31 (1,2): 1 (2,0): 231 (2,1): 31 (3,0): 234 (3,1): 1102 (3,2): 231
trapping sets H_Z (1,1)×19 (2,0)×66 (3,0)×463 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 19 (1,2): 11 (1,3): 2 (2,0): 66 (2,1): 140 (2,2): 68 (2,3): 10 (3,0): 463 (3,1): 963 (3,2): 732 (3,3): 335 (3,4): 66

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/32-28-1.json
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

[[32,26,2]] — C-ADD-CHECK strict improvement of 32-28-1

Direction & hypothesis

Convert the parent (codes/32-28-1.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/32-28-1.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/32-28-1.json; add the recorded checks to obtain this entry.

Parity checks

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