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}, { "X": [ 23, 35, 83, 110, 132, 156 ], "Z": [ 13, 35, 90, 113, 125, 184 ] }, { "X": [ 24, 36, 84, 111, 133, 157 ], "Z": [ 12, 34, 89, 112, 124, 183 ] }, { "X": [ 25, 37, 85, 112, 134, 158 ], "Z": [ 11, 33, 88, 111, 154, 182 ] }, { "X": [ 26, 38, 86, 113, 135, 159 ], "Z": [ 10, 32, 87, 110, 153, 181 ] }, { "X": [ 27, 39, 87, 114, 136, 160 ], "Z": [ 9, 31, 86, 109, 152, 180 ] }, { "X": [ 28, 40, 88, 115, 137, 161 ], "Z": [ 8, 61, 85, 108, 151, 179 ] } ] }, "distance": { "d": 13, "P": { "value": 13, "confidence": "upper_bound", "witness": { "X": [ 4, 8, 24, 58, 92, 95 ], "Z": [ 21, 28, 57, 92, 97, 101, 140, 152 ] } } }, "provenance": { "authors": [ "@vprusso" ], "origin": "submission", "construction": "search", "date": "2026-10-15", "references": [ "arXiv:2609.30069" ], "notes": "Halved CPM pair-partition code of arXiv:2609.30069 (Definition 1, reversing fold of Proposition 4) at (J,L)=(4,12), lift P=31. Exponent array E (rows a=0..3, columns l=0..11): [[7, 27, 2, 12, 4, 22, 15, 25, 19, 12, 6, 7], [21, 11, 15, 6, 26, 24, 3, 24, 6, 10, 22, 21], [12, 1, 21, 5, 3, 27, 0, 26, 19, 14, 3, 1], [2, 7, 21, 29, 4, 22, 8, 14, 17, 26, 24, 23]]. Off-diagonal pair partitions M: {\"0,1\": [[0, 9], [1, 5], [2, 6], [3, 10], [4, 7], [8, 11]], \"0,2\": [[0, 1], [2, 10], [3, 8], [4, 9], [5, 7], [6, 11]], \"0,3\": [[0, 8], [1, 9], [2, 11], [3, 7], [4, 5], [6, 10]], \"1,2\": [[0, 4], [1, 10], [2, 8], [3, 5], [6, 9], [7, 11]], \"1,3\": [[0, 5], [1, 2], [3, 9], [4, 6], [7, 8], [10, 11]], \"2,3\": [[0, 2], [1, 7], [3, 10], [4, 11], [5, 9], [6, 8]]}. The paper publishes no generators for its halved instances; this code is from a random search over the template. Distance 13 is the lightest Pauli logical found through 358 random per-qubit symplectic frames x 1,000,000 gf2_fast trials on the symplectic doubling (upper bound). The paper's Table VII lists [[186,65,14]] at this lift, as an exact value; this search did not reach 14, so the entry is 1 short of the published value. The same lift also carries [[186,65,15]] under the plain fold.", "model": "Claude Fable 5.1", "search_budget": { "candidates_screened": 16005, "ris_trials_per_side": 358000000, "tool": "halved_pp.py templates; verify/heuristic_distance.py Pauli RIS; gf2_fast on the symplectic doubling under random per-qubit symplectic frames (frame_screen.py)", "notes": "Pauli-weight RIS screen at 1200 trials per template (Perlmutter job 59705503, 2026-10-12), then 108 frames x 1,000,000 gf2_fast trials (jobs 59731365 to 59731368) and 250 further frames x 1,000,000 trials with fresh seeds (job 59779389, 2026-10-14). Lightest Pauli logical found: 13." } } }