[[76,43,2]] · QEC Challenge

[[76,43,2]] d ≤

kd²/n

2.26316

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 8, w_Z = 8 (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)

[68, 69]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[68, 69]

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 8 · H_Z 8

qubit degrees H_X 1–2 (mean 1.579) · H_Z 1–2 (mean 1.895)

trapping sets H_X (1,1)×32 (2,0)×162 (3,1)×352 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 32 (1,2): 44 (2,0): 162 (2,1): 32 (2,2): 160 (3,1): 352 (3,2): 1148

trapping sets H_Z (1,1)×8 (2,0)×114 (3,1)×184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 68 (2,0): 114 (2,1): 32 (2,2): 256 (3,1): 184 (3,2): 1844

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction code-calculus v0.2rc C-DELETE-CHECK

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

[[76,43,2]] greedy (target from 76-40-2

Direction & hypothesis

Unrestricted-weight CSS parents with 61 <= n <= 100, searched with the priced construction calculus (code-calculus v0.2rc) greedy multi-step composition. The rule used on this step is C-DELETE-CHECK.

What was searched

Greedy multi-step composition of C-DELETE-CHECK / E-PEEL / E-PUNCTURE with sector-targeted planning, sector swap, bounded support search, and Pareto ranking. This candidate is step (target of a chain from codes/76-40-2.json (target sector X).

Evidence trail

Staged deep-verify (2k -> 20k -> 200k -> 2M trials, multi-seed) kept the screened distance unchanged; the witness-backed upper bound is d_X <= 2, d_Z <= 2. No downgrade at any depth.

Dead ends

Parents whose display carries a dependent generator produce no derivable child; candidates whose deep-verified slug collides with an existing board entry are skipped.

Tools

Model: Beam 1.0. Harness: code-calculus v0.2rc greedy gated campaign, with the verifier's staged deep-verify used before submission.

Reproduction

Start from codes/76-40-2.json, apply C-DELETE-CHECK on the target sector, and repeat greedily with Pareto ranking and staged deep-verify.

Parity checks

X-checks 15 (max weight 8) · Z-checks 18 (max weight 8)

H_X (15 checks, sparse supports)

[0, 1, 36, 37, 38, 39, 74, 75] [2, 3, 4, 5, 40, 41, 42, 43] [6, 7, 8, 9, 44, 45, 46, 47] [10, 11, 12, 13, 48, 49, 50, 51] [12, 13, 14, 15, 50, 51, 52, 53] [14, 15, 16, 17, 52, 53, 54, 55] [16, 17, 18, 19, 54, 55, 56, 57] [18, 19, 20, 21, 56, 57, 58, 59] [20, 21, 22, 23, 58, 59, 60, 61] [22, 23, 24, 25, 60, 61, 62, 63] [24, 25, 26, 27, 62, 63, 64, 65] [26, 27, 28, 29, 64, 65, 66, 67] [28, 29, 30, 31, 66, 67, 68, 69] [30, 31, 32, 33, 68, 69, 70, 71] [32, 33, 34, 35, 70, 71, 72, 73]

H_Z (18 checks, sparse supports)

[0, 1, 2, 3, 38, 39, 40, 41] [2, 3, 4, 5, 40, 41, 42, 43] [4, 5, 6, 7, 42, 43, 44, 45] [6, 7, 8, 9, 44, 45, 46, 47] [8, 9, 10, 11, 46, 47, 48, 49] [10, 11, 12, 13, 48, 49, 50, 51] [12, 13, 14, 15, 50, 51, 52, 53] [14, 15, 16, 17, 52, 53, 54, 55] [16, 17, 18, 19, 54, 55, 56, 57] [18, 19, 20, 21, 56, 57, 58, 59] [20, 21, 22, 23, 58, 59, 60, 61] [22, 23, 24, 25, 60, 61, 62, 63] [24, 25, 26, 27, 62, 63, 64, 65] [26, 27, 28, 29, 64, 65, 66, 67] [28, 29, 30, 31, 66, 67, 68, 69] [30, 31, 32, 33, 68, 69, 70, 71] [32, 33, 34, 35, 70, 71, 72, 73] [34, 35, 36, 37, 72, 73, 74, 75]

Code ID 76-43-2 · download JSON · raw on GitHub