[[83,10,8]] · QEC Challenge

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[[83,10,8]] d ≤

kd²/n

7.71084

X/Z

1.12

0.0092

5.3852

layers

swaps

319

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Distance

X/Z asymmetry 1.12 · d_X ≤ 9, d_Z ≤ 8 · 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 9 · witness weight 9 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 9 qubits)

[5, 10, 20, 25, 40, 42, 51, 58, 68]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 8 qubits)

[2, 18, 23, 33, 35, 45, 49, 55]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 7 · H_Z 6–7 (mean 6.919)

qubit degrees H_X 2–4 (mean 3.036) · H_Z 1–4 (mean 3.084)

trapping sets H_X (1,2)×19 (2,2)×16 (3,1)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 19 (1,3): 42 (1,4): 22 (2,2): 16 (2,3): 115 (2,4): 255 (2,5): 293 (2,6): 77 (3,1): 4 (3,2): 49 (3,3): 278 (3,4): 1109 (3,5): 2429 (3,6): 3290 (3,7): 2105 (3,8): 643 (3,9): 244 (3,10): 27

trapping sets H_Z (1,1)×1 (2,2)×14 (3,1)×4 (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): 15 (1,3): 43 (1,4): 24 (2,2): 14 (2,3): 91 (2,4): 241 (2,5): 325 (2,6): 88 (3,1): 4 (3,2): 42 (3,3): 211 (3,4): 899 (3,5): 2218 (3,6): 3516 (3,7): 2361 (3,8): 735 (3,9): 287 (3,10): 33

witness diameter X 6.7082 · Z 5.099 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (42)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 319 nearest-neighbor SWAPs per round in total, at most 10 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus E-PEEL of codes/84-10-9.json ([[84,10,9]], max check weight 7; a generalized-bicycle entry by @MathysRennela): delete qubit 38 (0-based index in the parent file) together with its unique X check a8, 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)

date 2026-11-04

notes Single-step priced-calculus E-PEEL child of board entry 84-10-9. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.

family generalized bicycle (a tag, not a ranking)

locality 2D-local bilayer (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

[[83,10,8]] priced calculus E-PEEL of codes/84-10-9.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/84-10-9.json ([[84,10,9]], max check weight 7; a generalized-bicycle entry by @MathysRennela). Peeling qubit 38 with the one X check that touches it keeps k = 10 and lowers the witnessed distance bound by one, from 9 to 8, while dropping one qubit. Max check weight stays 7. The layout is the parent's with the peeled qubit removed; the verifier re-measured its radius and layer count.

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 2,000 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 [[83,10,8]] 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 | | ---: | ---: | | 2,000 (staging screen, seeds 1 and 2) | 8 | | 1,000,000 (seed 101) | 8 | | 1,000,000 (seed 102) | 8 | | 10,000,000 (seed 103) | 8 |

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 <= 9 (upper_bound), Z: d <= 8 (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/84-10-9.json, delete qubit 38 (0-based index in that file) together with its unique X check a8, where the X rows are numbered a1, a2, ... 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': 38, 'sector': 'X', 'row_id': 'a8'}). 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 36 (max weight 7) · Z-checks 37 (max weight 7)

H_X (36 checks, sparse supports)

[0, 28, 39, 41, 54, 75, 76] [1, 25, 40, 42, 57, 72, 73] [2, 32, 33, 43, 59, 79, 80] [3, 6, 34, 42, 44, 60, 80] [4, 30, 35, 45, 62, 77, 78] [5, 9, 31, 41, 46, 63, 78] [6, 23, 37, 47, 49, 70, 82] [8, 12, 38, 45, 46, 49, 66] [9, 20, 38, 50, 52, 67, 81] [1, 10, 36, 43, 51, 68, 81] [11, 15, 37, 43, 44, 52, 69] [2, 12, 29, 50, 53, 55, 76] [3, 13, 40, 50, 51, 54, 71] [4, 14, 18, 51, 52, 55, 72] [4, 15, 26, 47, 56, 58, 73] [5, 16, 39, 47, 48, 57, 74] [2, 17, 21, 48, 49, 58, 75] [7, 18, 35, 42, 56, 59, 61] [8, 9, 19, 56, 57, 60, 77] [10, 20, 24, 57, 58, 61, 78] [10, 21, 32, 41, 53, 62, 64] [6, 11, 22, 53, 54, 63, 79] [7, 23, 27, 54, 55, 64, 80] [1, 13, 24, 46, 62, 65, 67] [14, 15, 25, 62, 63, 66, 81] [16, 26, 30, 43, 63, 64, 67] [0, 16, 27, 44, 59, 68, 70] [12, 17, 28, 59, 60, 69, 82] [13, 29, 33, 45, 60, 61, 70] [5, 19, 30, 52, 68, 71, 73] [20, 21, 31, 47, 68, 69, 72] [22, 32, 36, 48, 69, 70, 73] [18, 23, 34, 50, 65, 66, 75] [11, 25, 36, 41, 58, 74, 77] [26, 27, 37, 53, 74, 75, 78] [17, 31, 39, 44, 64, 79, 81]

H_Z (37 checks, sparse supports)

[0, 5, 21, 36, 41, 48, 68] [1, 3, 18, 42, 51, 65] [2, 10, 11, 26, 43, 53, 58] [3, 11, 27, 39, 44, 54, 74] [4, 7, 8, 29, 45, 55, 56] [5, 8, 24, 40, 46, 57, 71] [6, 15, 16, 31, 44, 47, 63] [7, 16, 17, 32, 48, 59, 64] [6, 8, 17, 33, 49, 60, 79] [9, 12, 13, 34, 46, 50, 60] [10, 13, 14, 35, 51, 61, 62] [9, 11, 14, 30, 52, 63, 77] [12, 21, 22, 37, 49, 53, 69] [0, 13, 22, 23, 54, 65, 70] [12, 14, 23, 55, 66, 82] [15, 18, 19, 38, 52, 56, 66] [1, 16, 19, 20, 57, 67, 68] [15, 17, 20, 36, 58, 69, 81] [2, 18, 27, 28, 55, 59, 75] [3, 19, 28, 29, 60, 71, 76] [18, 20, 29, 40, 50, 61, 72] [4, 21, 24, 25, 58, 62, 72] [5, 22, 25, 26, 63, 73, 74] [21, 23, 26, 39, 47, 64, 75] [7, 24, 33, 34, 61, 65, 80] [8, 25, 34, 35, 42, 66, 77] [9, 24, 26, 35, 56, 67, 78] [10, 27, 30, 31, 64, 68, 78] [11, 28, 31, 32, 41, 69, 79] [6, 27, 29, 32, 53, 70, 80] [13, 30, 38, 45, 67, 71] [1, 14, 31, 38, 46, 72, 81] [1, 15, 30, 32, 43, 62, 73] [0, 17, 34, 37, 44, 75, 82] [4, 19, 36, 40, 51, 73, 77] [4, 5, 20, 37, 47, 52, 78] [9, 10, 25, 39, 41, 57, 81]

Code ID 83-10-8 · download JSON · raw on GitHub