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64, 68, 101, 109, 123, 170, 235, 276, 351, 353, 436, 450, 472, 604, 613, 642 ], [ 65, 69, 102, 110, 124, 171, 236, 277, 352, 354, 437, 451, 473, 605, 614, 643 ], [ 66, 70, 103, 111, 125, 172, 237, 278, 353, 355, 438, 452, 474, 606, 615, 644 ], [ 67, 71, 104, 112, 126, 173, 238, 279, 354, 356, 439, 453, 475, 607, 616, 645 ], [ 68, 72, 105, 113, 127, 174, 239, 280, 355, 357, 440, 454, 476, 608, 617, 646 ], [ 69, 73, 106, 114, 128, 175, 240, 281, 356, 358, 441, 455, 477, 609, 618, 647 ], [ 70, 74, 107, 115, 129, 176, 241, 282, 357, 359, 442, 456, 478, 610, 619, 648 ], [ 71, 75, 108, 116, 130, 177, 242, 283, 358, 360, 443, 457, 479, 611, 620, 649 ] ] }, "family": "generalized-bicycle", "provenance": { "authors": [ "@vprusso" ], "model": "Claude Opus 5", "novelty": "known_parameters", "origin": "submission", "date": "2026-10-13", "construction": "Cyclic generalized-bicycle code on the ring x^341 - 1 with a(x) = x^32 + x^61 + x^70 + x^202 + x^224 + x^238 + x^321 + x^323, b(x) = x^57 + x^98 + x^163 + x^210 + x^224 + x^232 + x^265 + x^269; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]. a and b are low-weight codewords of the ideal of a divisor of x^341 - 1 of degree 51, fixing k = 2 deg gcd(a, b, x^m - 1) = 102.", "notes": "Designed-divisor construction: the divisor is a product of cyclotomic-coset minimal polynomials over GF(2^ord_2(m)), so its degree and hence k are chosen rather than inherited, and a and b are drawn from its ideal by information-set decoding. Both single-block kernels have dimension 51 exactly, matching deg gcd(a, b, x^m - 1). The binding bound here is the orbit fold, not the general search: a 24-seed information-set gate at 2,000,000 trials each over 16,105 s bottoms at 86 with a per-seed spread of 86 to 93, while the fold over every equal-orbit partition of the 682 qubits returns a validated lift of weight 77 at 31 orbits of size 11. The claim is therefore 77 on X from the fold and 86 on Z from the gate. A single-block circulant search at 10 seeds x 600,000 trials finds nothing under 77. Both witnesses re-verified against these matrices." } }