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[[34,7,2]] d ≤
n
34
k
7
d
2
kd²/n
0.823529
w
4
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 4, w_Z = 4 (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, 7]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 6]
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 acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.5) · H_Z 2–4 (mean 3.2)
qubit degrees H_X 1–2 (mean 1.235) · H_Z 1–2 (mean 1.412)
trapping sets H_X (1,1)×26 (2,0)×19 (3,0)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 26 (1,2): 8 (2,0): 19 (2,1): 32 (2,2): 6 (3,0): 32 (3,1): 30 (3,2): 22 (3,3): 12
trapping sets H_Z (1,1)×20 (2,0)×10 (3,0)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 20 (1,2): 14 (2,0): 10 (2,1): 34 (2,2): 16 (3,0): 18 (3,1): 42 (3,2): 28 (3,3): 17 (3,4): 3

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/34-9-1.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 ≤ 4 (computed)

How this code was found

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

[[34,7,2]] — C-ADD-CHECK strict improvement of 34-9-1

Direction & hypothesis

Convert the parent (codes/34-9-1.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/34-9-1.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/34-9-1.json; add the recorded checks to obtain this entry.

Parity checks

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