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76, 112, 268, 320, 355, 439, 538 ], [ 73, 77, 113, 269, 321, 354, 438, 539 ] ] }, "distance": { "X": { "value": 41, "confidence": "upper_bound", "witness": [ 5, 9, 15, 19, 21, 23, 51, 61, 63, 85, 89, 97, 99, 103, 119, 123, 143, 161, 163, 165, 175, 177, 179, 183, 197, 205, 241, 245, 257, 278, 317, 328, 362, 363, 397, 415, 416, 449, 500, 514, 535 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-26", "found_at_samples": 1000000000, "tool": "verify/ris_gpu.py (build/ris_gpu, NVIDIA A40)", "seeds": [ 1391721037 ], "survived_samples": 1000000000 } }, "Z": { "value": 44, "confidence": "upper_bound", "witness": [ 13, 14, 15, 32, 65, 67, 76, 78, 82, 97, 98, 100, 114, 117, 128, 130, 133, 135, 148, 149, 150, 151, 163, 165, 166, 167, 182, 185, 196, 200, 219, 232, 234, 235, 247, 249, 250, 266, 274, 302, 306, 376, 466, 502 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-26", "found_at_samples": 50000000, "tool": "verify/ris_gpu.py (build/ris_gpu, NVIDIA A40)", "seeds": [ 1391721033, 1391721034, 1391721035, 1391721036, 1391721037, 1391721038 ], "survived_samples": 300000000 } }, "d": 41 }, "provenance": { "authors": [ "@vprusso" ], "construction": "Periodic bivariate-bicycle code on Z_135 x Z_2 (n = 2*l*m = 540): x = S_135 tensor I_2, y = I_135 tensor S_2 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^50y^1 + x^92y^1 + x^109y^0, B = x^0y^0 + x^78y^0 + x^96y^0 + x^98y^0; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(135, 2, [[0, 0], [50, 1], [92, 1], [109, 0]], [[0, 0], [78, 0], [96, 0], [98, 0]])). gcd(135, 2) = 1, so Z_135 x Z_2 is cyclic of order 270 and the code is the cyclic generalized-bicycle code over Z_270 with a(z) = z^0 + z^185 + z^227 + z^244, b(z) = z^0 + z^78 + z^96 + z^98 (CRT relabeling x^a y^b -> z^t, t = a mod 135, t = b mod 2).", "date": "2026-09-26", "notes": "Found by the fourth LER-objective search (coprime bivariate-bicycle codes (gcd(l, m) = 1) with check weight 6, 7, or 8 at 420 < n <= 700 and k >= 8): circuit-level logical error rate at 3 rounds under the board's depolarizing recipe, interleaved schedule, decoded with CUDA-Q QEC nv-qldpc-decoder (BP+OSD-CS order 10). At p = 0.002, 100000 shots per basis: Z memory 11 failures, X memory 13 failures, against 12 copies of the distance-7 rotated surface code (1164 physical qubits, same builder and decoder) with 101 and 88 failures. Decoder-based d_circ estimate at 3 rounds: 64. Distance: RIS at 20,000 trials (gf2_fast) and a 300,000,000-trial verify/ris_gpu.py pass per side; upper bounds. A norm-lift quotient bound over the divisors of m and a single-block kernel bound were run before admission (lightest lift 45). Details in the accompanying note.", "model": "Claude Fable 5.1 (Claude Code)", "origin": "submission", "novelty": "unknown" } }