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[[39,25,2]] d ≤
n
39
k
25
d
2
kd²/n
2.5641
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 10, w_Z = 10 (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)
[1, 27]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[3, 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 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 10 · H_Z 9–10 (mean 9.833)
qubit degrees H_X 1–4 (mean 2.051) · H_Z 1–3 (mean 1.513)
trapping sets H_X (1,1)×11 (2,0)×6 (3,0)×69 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 11 (1,2): 18 (1,3): 7 (1,4): 3 (2,0): 6 (2,1): 59 (2,2): 106 (2,3): 94 (2,4): 42 (2,5): 10 (3,0): 69 (3,1): 370 (3,2): 768 (3,3): 919 (3,4): 663 (3,5): 313 (3,6): 84 (3,7): 16
trapping sets H_Z (1,1)×24 (2,0)×47 (3,0)×146 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 24 (1,2): 10 (1,3): 5 (2,0): 47 (2,1): 74 (2,2): 86 (2,3): 32 (2,4): 6 (3,0): 146 (3,1): 524 (3,2): 621 (3,3): 421 (3,4): 165 (3,5): 47 (3,6): 4

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction C-ADD-CHECK strict improvement (code calculus v0.2rc): added one check killing every minimum-weight logical of parent codes/39-26-1.json
model Beam 1.0 (claimed, not verified)
date 2026-10-29
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

[[39,25,2]] — C-ADD-CHECK strict improvement of 39-26-1

Direction & hypothesis

Convert the parent (codes/39-26-1.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 check, 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 support is 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/39-26-1.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/39-26-1.json; add the recorded check to obtain this entry.

Parity checks

X-checks 8 (max weight 10) · Z-checks 6 (max weight 10)
H_X (8 checks, sparse supports)
[0, 1, 15, 17, 20, 22, 23, 24, 27, 30] [0, 1, 16, 18, 19, 21, 25, 26, 27, 30] [0, 2, 3, 6, 23, 25, 27, 28, 32, 33] [4, 6, 7, 10, 27, 29, 31, 32, 36, 37] [3, 5, 8, 9, 28, 30, 31, 32, 35, 38] [8, 10, 11, 14, 20, 21, 31, 33, 35, 36] [7, 9, 12, 13, 19, 22, 32, 34, 35, 36] [12, 14, 15, 18, 19, 20, 24, 25, 35, 37]
H_Z (6 checks, sparse supports)
[0, 2, 11, 14, 15, 16, 20, 21, 23, 25] [0, 1, 3, 4, 7, 9, 27, 30, 32, 34] [4, 5, 7, 8, 11, 13, 31, 34, 36, 38] [3, 6, 7, 8, 12, 14, 32, 33, 35, 37] [8, 9, 11, 12, 15, 17, 20, 22, 35, 38] [0, 10, 18, 19, 24, 26, 28, 29, 35]
Code ID 39-25-2 · download JSON · raw on GitHub