[[40,6,6]] · QEC Challenge

[[40,6,6]] d ≤

kd²/n

5.4

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-11-03

witness operator (support, 6 qubits)

[8, 13, 18, 23, 30, 38]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-11-03

witness operator (support, 6 qubits)

[0, 10, 13, 17, 19, 36]

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 6–7 (mean 6.882) · H_Z 6–7 (mean 6.882)

qubit degrees H_X 1–4 (mean 2.925) · H_Z 1–4 (mean 2.925)

trapping sets H_X (1,1)×1 (2,1)×4 (3,1)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 10 (1,3): 20 (1,4): 9 (2,1): 4 (2,2): 26 (2,3): 72 (2,4): 108 (2,5): 79 (2,6): 14 (3,1): 15 (3,2): 89 (3,3): 343 (3,4): 643 (3,5): 787 (3,6): 681 (3,7): 304 (3,8): 84 (3,9): 19

trapping sets H_Z (1,1)×1 (2,1)×4 (3,1)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 10 (1,3): 20 (1,4): 9 (2,1): 4 (2,2): 26 (2,3): 72 (2,4): 108 (2,5): 79 (2,6): 14 (3,1): 15 (3,2): 89 (3,3): 343 (3,4): 643 (3,5): 787 (3,6): 681 (3,7): 304 (3,8): 84 (3,9): 19

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus E-PEEL of codes/41-6-6.json ([[41,6,6]], max check weight 7; itself a priced-calculus E-PEEL of codes/42-6-7.json, a Lin and Pryadko 2BGA code over SmallGroup(21,2)): delete qubit 40 (0-based index in the parent file) together with its unique Z check b11, and relabel the surviving qubits in increasing order of their parent indices. Consumer: github.com/unitaryfoundation/code-calculus @ 1b9a942, machine/priced_calculus.py.

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2306.16400

date 2026-11-04

notes Single-step priced-calculus E-PEEL child of board entry 41-6-6. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.

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

[[40,6,6]] priced calculus E-PEEL of codes/41-6-6.json

Direction & hypothesis

Unrestricted weight-8 CSS cell. The priced construction calculus was applied one move deep to every CSS entry on the board, and this code is the E-PEEL of codes/41-6-6.json ([[41,6,6]], max check weight 7; itself a priced-calculus E-PEEL of codes/42-6-7.json, a Lin and Pryadko 2BGA code over SmallGroup(21,2)). Peeling qubit 40 with the one Z check that touches it keeps k = 6 and the witnessed distance bound d <= 6 while dropping one qubit, so it sits one qubit to the left of its parent on the frontier. Max check weight stays 7.

What was searched

Every C-DELETE-CHECK, E-PEEL, and E-PUNCTURE move on each of the 2,783 CSS board entries as of 2026-11-02, 1,684,787 moves in all, with the consumer at github.com/unitaryfoundation/code-calculus @ 1b9a942. Children with a bare qubit, a disconnected Tanner graph, a Weisfeiler-Leman signature equal to a board entry or to an earlier child, or parameters already dominated inside the parent's cell were dropped before any search, leaving 334,191 distinct children. Children with n <= 60 and an inherited distance bound of at least 2, and children with 61 <= n <= 100 and a bound of at least 5, were screened with research/kit/surrogate.lightest_logical at 3,000 RIS trials per side; the survivors the kit's novelty pre-check called board-advancing were staged with research/kit/submit.py and gated with verify/validate_candidate.py, one representative per distinct (n, k, d, max weight, locality) point. This code is that representative for [[40,6,6]] at max check weight 7. The screen distance is a witness-backed upper bound; the rungs below are the independent confirmation runs on the staged file.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 3,000 (staging screen, seeds 1 and 2) | 6 | | 1,000,000 (seed 101) | 6 | | 1,000,000 (seed 102) | 6 | | 10,000,000 (seed 103) | 6 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 10,000,000 trials per side. The submitted witness reproduces it: X: d <= 6 (upper_bound), Z: d <= 6 (upper_bound).

Dead ends

None recorded for this candidate.

Tools

Model: Claude Fable 5.1 (Claude Code). Repo tooling: research/kit/coordination.py, research/kit/submit.py, research/kit/surrogate.py, research/audits/leader_audit.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py.

Reproduction

Load codes/41-6-6.json, delete qubit 40 (0-based index in that file) together with its unique Z check b11, where the Z rows are numbered b1, b2, ... in file order after dropping rows dependent on earlier ones, as CodeState.from_challenge in machine/priced_calculus.py labels them, and relabel the surviving qubits in increasing order of their parent indices; in github.com/unitaryfoundation/code-calculus @ 1b9a942 this is apply_operation(state, {'op': 'E-PEEL', 'qubit': 40, 'sector': 'Z', 'row_id': 'b11'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 17 (max weight 7) · Z-checks 17 (max weight 7)

H_X (17 checks, sparse supports)

[1, 11, 18, 21, 22, 32, 37] [0, 2, 17, 22, 23, 29, 33] [3, 13, 19, 23, 24, 28, 39] [1, 4, 14, 20, 24, 25, 35] [2, 5, 19, 25, 26, 32, 36] [3, 6, 15, 21, 26, 27, 31] [4, 7, 16, 22, 27, 28, 38] [3, 5, 8, 28, 29, 35, 39] [6, 9, 18, 24, 29, 30, 34] [0, 7, 10, 25, 30, 31] [6, 8, 11, 21, 31, 32, 38] [1, 9, 12, 27, 32, 33, 37] [2, 10, 13, 23, 28, 33, 34] [9, 11, 14, 24, 34, 35] [5, 13, 15, 26, 31, 36, 37] [12, 14, 16, 23, 27, 37, 38] [8, 15, 18, 20, 29, 34, 39]

H_Z (17 checks, sparse supports)

[0, 4, 18, 20, 22, 30] [1, 6, 11, 19, 21, 24, 32] [1, 2, 7, 17, 22, 25, 33] [2, 3, 13, 16, 23, 26, 28] [3, 4, 9, 14, 24, 27, 35] [4, 5, 10, 19, 25, 28, 36] [0, 5, 6, 15, 26, 29, 31] [6, 7, 12, 16, 27, 30, 38] [3, 7, 8, 13, 28, 31, 39] [2, 8, 9, 18, 29, 32, 34] [6, 10, 11, 15, 21, 31, 34] [1, 5, 11, 12, 32, 35, 37] [2, 12, 13, 17, 23, 33, 36] [9, 13, 14, 18, 24, 34, 37] [5, 15, 19, 26, 36, 39] [1, 12, 15, 16, 20, 27, 37] [3, 8, 17, 18, 21, 29, 39]

Code ID 40-6-6-b · download JSON · raw on GitHub