[[96,31,8]] · QEC Challenge

[[96,31,8]] d ≤

kd²/n

20.6667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 32, w_Z = 32 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 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)

[1, 2, 41, 42, 49, 50, 89, 90]

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)

[0, 2, 44, 46, 48, 50, 92, 94]

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 8–32 (mean 22.125) · H_Z 8–32 (mean 22.545)

qubit degrees H_X 1–18 (mean 7.375) · H_Z 2–17 (mean 7.75)

trapping sets H_X (1,1)×1 (2,1)×5 (3,1)×5 (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): 4 (1,3): 9 (1,4): 15 (1,5): 12 (1,6): 6 (1,7): 4 (1,8): 5 (1,9): 5 (1,10): 10 (1,11): 13 (1,12): 5 (1,13): 3 (1,14): 2 (1,15): 1 (1,18): 1 (2,1): 5 (2,2): 17 (2,3): 51 (2,4): 81 (2,5): 120 (2,6): 133 (2,7): 169 (2,8): 187 (2,9): 230 (2,10): 331 (2,11): 410 (2,12): 405 (2,13): 425 (2,14): 297 (2,15): 182 (2,16): 143 (2,17): 58 (2,18): 35 (2,19): 21 (2,20): 12 (2,21): 1 (3,1): 5 (3,2): 51 (3,3): 180 (3,4): 419 (3,5): 932 (3,6): 1654 (3,7): 2525 (3,8): 3801 (3,9): 5700 (3,10): 8296 (3,11): 11247 (3,12): 13906 (3,13): 14979 (3,14): 14269 (3,15): 11712 (3,16): 8285 (3,17): 5342 (3,18): 3085 (3,19): 1591 (3,20): 732 (3,21): 292 (3,22): 107 (3,23): 22 (3,24): 6 (3,25): 1

trapping sets H_Z (1,2)×5 (2,1)×3 (3,1)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 5 (1,3): 7 (1,4): 15 (1,5): 10 (1,6): 7 (1,7): 5 (1,8): 4 (1,9): 6 (1,10): 13 (1,11): 5 (1,12): 8 (1,13): 4 (1,14): 3 (1,16): 3 (1,17): 1 (2,1): 3 (2,2): 9 (2,3): 36 (2,4): 64 (2,5): 103 (2,6): 132 (2,7): 159 (2,8): 195 (2,9): 241 (2,10): 305 (2,11): 351 (2,12): 452 (2,13): 394 (2,14): 322 (2,15): 237 (2,16): 187 (2,17): 97 (2,18): 69 (2,19): 41 (2,20): 13 (2,21): 3 (2,22): 2 (3,1): 5 (3,2): 33 (3,3): 110 (3,4): 297 (3,5): 654 (3,6): 1208 (3,7): 2227 (3,8): 3502 (3,9): 5372 (3,10): 7596 (3,11): 10317 (3,12): 13022 (3,13): 14614 (3,14): 14687 (3,15): 12928 (3,16): 10174 (3,17): 7068 (3,18): 4480 (3,19): 2517 (3,20): 1219 (3,21): 541 (3,22): 209 (3,23): 55 (3,24): 12 (3,25): 1 (3,26): 1

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus C-DELETE-CHECK of codes/96-30-8.json ([[96,30,8]], max check weight 32; the chain code q96_30_8 of Hirai and Suzuki, arXiv:2610.04613, with a weight-reduced generating set): delete X check a21 (rows numbered in file order after dropping dependent rows, as machine/priced_calculus.py labels them) and keep every qubit at its parent index. Consumer: github.com/unitaryfoundation/code-calculus @ 1b9a942, machine/priced_calculus.py.

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2610.04613

date 2026-11-04

notes Single-step priced-calculus C-DELETE-CHECK child of board entry 96-30-8. 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

[[96,31,8]] priced calculus C-DELETE-CHECK of codes/96-30-8.json

Direction & hypothesis

Unrestricted weight-9plus CSS cell. The priced construction calculus was applied one move deep to every CSS entry on the board, and this code is the C-DELETE-CHECK of codes/96-30-8.json ([[96,30,8]], max check weight 32; the chain code q96_30_8 of Hirai and Suzuki, arXiv:2610.04613, with a weight-reduced generating set). Deleting X check a21 raises k from 30 to 31 and keeps the witnessed distance bound at 8; n = 96 and the max check weight 32 are unchanged.

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 [[96,31,8]] at max check weight 32. The thirteen points audited first went to the board on 2026-11-04 (PRs #4588 to #4600); this one is from the remainder, re-checked against the board of 2026-11-04 with the gate's structural, deduplication, and novelty stages before the ladder below was run. 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 <= 8 (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/96-30-8.json and delete X check a21, 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; every qubit keeps its index. In github.com/unitaryfoundation/code-calculus @ 1b9a942 this is apply_operation(state, {'op': 'C-DELETE-CHECK', 'row_id': 'a21'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 10000000:103 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 32 (max weight 32) · Z-checks 33 (max weight 32)

H_X (32 checks, sparse supports)

[3, 6, 20, 22, 25, 27, 29, 31, 33, 34, 43, 47, 51, 54, 68, 70, 73, 75, 77, 79, 81, 82, 91, 95] [40, 41, 42, 43, 44, 45, 46, 47] [5, 7, 8, 9, 11, 12, 13, 15, 17, 22, 25, 26, 28, 29, 33, 37, 53, 55, 58, 62, 65, 70, 73, 74, 76, 77, 81, 85] [48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94] [32, 33, 34, 35, 36, 37, 38, 39] [48, 51, 53, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 75, 77, 78, 80, 83, 85, 86, 88, 91, 93, 94] [1, 3, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 80, 81, 82, 83, 84, 85, 86, 87] [3, 5, 11, 12, 13, 14, 18, 19, 26, 31, 41, 47, 51, 53, 59, 60, 61, 62, 66, 67, 74, 79, 89, 95] [64, 65, 66, 67, 68, 69, 70, 71] [56, 57, 58, 59, 60, 61, 62, 63] [48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 82, 83, 86, 87, 88, 89, 92, 93] [3, 6, 10, 11, 12, 14, 17, 19, 25, 29, 41, 44, 45, 47, 51, 54, 58, 59, 60, 62, 65, 67, 73, 77, 89, 92, 93, 95] [5, 7, 9, 10, 11, 13, 14, 15, 20, 22, 25, 31, 33, 37, 41, 47, 53, 55, 57, 58, 59, 61, 62, 63, 68, 70, 73, 79, 81, 85, 89, 95] [3, 5, 10, 12, 20, 22, 25, 28, 29, 31, 33, 35, 51, 53, 58, 60, 68, 70, 73, 76, 77, 79, 81, 83] [6, 7, 10, 11, 17, 19, 25, 31, 37, 38, 43, 44, 54, 55, 58, 59, 65, 67, 73, 79, 85, 86, 91, 92] [5, 6, 9, 12, 14, 15, 17, 22, 25, 29, 33, 35, 42, 43, 53, 54, 57, 60, 62, 63, 65, 70, 73, 77, 81, 83, 90, 91] [9, 15, 17, 20, 25, 26, 36, 37, 41, 43, 44, 47, 57, 63, 65, 68, 73, 74, 84, 85, 89, 91, 92, 95] [3, 6, 9, 10, 11, 12, 13, 14, 17, 19, 20, 22, 29, 31, 37, 39, 41, 43, 51, 54, 56, 63, 65, 67, 68, 70, 77, 79, 85, 87, 89, 91] [9, 10, 11, 13, 19, 20, 21, 22, 26, 31, 43, 47, 57, 58, 59, 61, 67, 68, 69, 70, 74, 79, 91, 95] [11, 12, 25, 26, 29, 31, 43, 44, 46, 47, 59, 60, 73, 74, 77, 79, 91, 92, 94, 95] [11, 13, 19, 20, 25, 26, 27, 31, 33, 37, 41, 44, 59, 61, 67, 68, 73, 74, 75, 79, 81, 85, 89, 92] [3, 7, 9, 10, 12, 13, 17, 19, 20, 23, 25, 26, 44, 47, 51, 55, 56, 59, 62, 63, 65, 67, 68, 71, 73, 74, 92, 95] [16, 17, 18, 19, 20, 21, 22, 23] [24, 25, 26, 27, 28, 29, 30, 31, 64, 65, 66, 67, 68, 69, 70, 71] [0, 3, 5, 6, 8, 11, 13, 14, 16, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 38, 40, 43, 45, 46] [0, 1, 2, 3, 4, 5, 6, 7] [72, 73, 74, 75, 76, 77, 78, 79] [48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93] [0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 26, 27, 30, 31, 32, 33, 36, 37, 40, 41, 44, 45, 72, 73, 74, 75, 76, 77, 78, 79] [3, 6, 10, 12, 13, 14, 25, 31, 41, 42, 44, 46, 48, 49, 50, 52, 53, 55, 58, 60, 61, 62, 73, 79, 89, 90, 92, 94] [5, 6, 10, 13, 17, 18, 25, 26, 29, 31, 44, 45, 53, 54, 58, 61, 65, 66, 73, 74, 77, 79, 88, 89, 90, 91, 94, 95] [3, 7, 8, 9, 10, 11, 13, 15, 17, 19, 20, 22, 25, 26, 29, 30, 41, 47, 51, 55, 60, 62, 65, 67, 68, 70, 73, 74, 77, 78, 89, 95]

H_Z (33 checks, sparse supports)

[1, 4, 20, 22, 24, 27, 29, 31, 32, 34, 43, 47, 49, 52, 68, 70, 72, 75, 77, 79, 80, 82, 91, 95] [40, 41, 42, 43, 44, 45, 46, 47] [0, 1, 2, 7, 8, 9, 11, 12, 13, 15, 16, 22, 24, 26, 28, 29, 32, 37, 48, 49, 50, 55, 58, 62, 64, 70, 72, 74, 76, 77, 80, 85] [50, 52, 54, 55, 58, 60, 62, 63, 66, 68, 70, 71, 74, 76, 78, 79, 82, 84, 86, 87, 90, 92, 94, 95] [32, 33, 34, 35, 36, 37, 38, 39] [51, 53, 54, 55, 59, 61, 62, 63, 67, 69, 70, 71, 75, 77, 78, 79, 83, 85, 86, 87, 91, 93, 94, 95] [0, 1, 3, 5, 10, 12, 14, 15, 18, 20, 22, 23, 26, 28, 30, 31, 34, 36, 38, 39, 40, 41, 43, 45, 80, 81, 82, 83, 84, 85, 86, 87] [0, 4, 8, 12, 13, 14, 18, 19, 26, 31, 40, 47, 48, 52, 56, 60, 61, 62, 66, 67, 74, 79, 88, 95] [64, 65, 66, 67, 68, 69, 70, 71] [56, 57, 58, 59, 60, 61, 62, 63] [49, 52, 53, 55, 57, 60, 61, 63, 65, 68, 69, 71, 73, 76, 77, 79, 80, 82, 83, 86, 89, 92, 93, 95] [8, 10, 12, 14, 16, 19, 24, 29, 40, 44, 45, 47, 56, 58, 60, 62, 64, 67, 72, 77, 88, 92, 93, 95] [0, 2, 4, 7, 10, 13, 14, 15, 20, 22, 24, 31, 32, 37, 40, 47, 48, 50, 52, 55, 58, 61, 62, 63, 68, 70, 72, 79, 80, 85, 88, 95] [0, 1, 10, 12, 20, 22, 24, 28, 29, 31, 32, 35, 48, 49, 58, 60, 68, 70, 72, 76, 77, 79, 80, 83] [0, 1, 4, 7, 8, 10, 16, 19, 24, 31, 37, 38, 43, 44, 48, 49, 52, 55, 56, 58, 64, 67, 72, 79, 85, 86, 91, 92] [0, 4, 8, 12, 14, 15, 16, 22, 24, 29, 32, 35, 42, 43, 48, 52, 56, 60, 62, 63, 64, 70, 72, 77, 80, 83, 90, 91] [8, 15, 16, 20, 24, 26, 36, 37, 40, 43, 44, 47, 56, 63, 64, 68, 72, 74, 84, 85, 88, 91, 92, 95] [0, 2, 10, 13, 19, 20, 21, 22, 26, 31, 43, 47, 48, 50, 58, 61, 67, 68, 69, 70, 74, 79, 91, 95] [0, 2, 8, 12, 24, 26, 29, 31, 43, 44, 46, 47, 48, 50, 56, 60, 72, 74, 77, 79, 91, 92, 94, 95] [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63] [1, 4, 8, 13, 19, 20, 24, 26, 27, 31, 32, 37, 40, 44, 49, 52, 56, 61, 67, 68, 72, 74, 75, 79, 80, 85, 88, 92] [0, 1, 4, 7, 8, 10, 12, 13, 16, 19, 20, 23, 24, 26, 44, 47, 48, 49, 52, 55, 57, 59, 62, 63, 64, 67, 68, 71, 72, 74, 92, 95] [16, 17, 18, 19, 20, 21, 22, 23] [24, 25, 26, 27, 28, 29, 30, 31, 64, 65, 66, 67, 68, 69, 70, 71] [3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 27, 29, 30, 31, 35, 37, 38, 39, 43, 45, 46, 47] [0, 1, 2, 3, 4, 5, 6, 7] [72, 73, 74, 75, 76, 77, 78, 79] [8, 10, 13, 14, 24, 25, 27, 28, 30, 31, 40, 42, 43, 47, 56, 58, 61, 62, 74, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 95] [49, 52, 53, 55, 57, 60, 61, 63, 65, 68, 69, 71, 73, 76, 77, 79, 81, 84, 85, 87, 89, 92, 93, 95] [1, 4, 5, 7, 9, 12, 13, 15, 17, 20, 21, 23, 24, 26, 27, 30, 33, 36, 37, 39, 41, 44, 45, 47, 72, 73, 74, 75, 76, 77, 78, 79] [0, 1, 2, 4, 8, 13, 20, 22, 24, 31, 37, 39, 40, 41, 42, 46, 48, 49, 50, 52, 56, 61, 68, 70, 72, 79, 85, 87, 91, 92, 93, 95] [0, 4, 10, 13, 16, 18, 24, 26, 29, 31, 44, 45, 48, 52, 58, 61, 64, 66, 72, 74, 77, 79, 88, 89, 90, 91, 94, 95] [2, 7, 8, 9, 10, 11, 13, 15, 16, 19, 20, 22, 24, 26, 29, 30, 40, 47, 50, 55, 60, 62, 64, 67, 68, 70, 72, 74, 77, 78, 88, 95]

Code ID 96-31-8 · download JSON · raw on GitHub