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41, 43, 50, 52, 54, 56, 58, 66, 67, 74, 120, 125, 185, 194, 228, 254, 298, 317, 321, 327, 362, 381, 390, 406, 438, 444, 455, 460, 481, 495, 502 ], [ 39, 42, 44, 51, 53, 55, 57, 59, 67, 68, 75, 121, 126, 186, 195, 229, 255, 299, 318, 322, 328, 363, 382, 391, 407, 439, 445, 456, 461, 482, 496, 503 ], [ 40, 43, 45, 52, 54, 56, 58, 60, 68, 69, 76, 122, 127, 187, 196, 230, 256, 300, 319, 323, 329, 364, 383, 392, 408, 440, 446, 457, 462, 483, 497, 504 ], [ 0, 41, 44, 46, 53, 55, 57, 59, 61, 69, 70, 77, 123, 128, 188, 197, 231, 301, 320, 324, 330, 365, 384, 393, 409, 441, 447, 458, 463, 484, 498, 505 ] ] }, "family": "generalized-bicycle", "provenance": { "authors": [ "@vprusso" ], "model": "Claude Opus 5", "novelty": "known_parameters", "origin": "submission", "date": "2026-10-12", "construction": "Cyclic generalized-bicycle code on the ring x^257 - 1 with a(x) = x^8 + x^15 + x^29 + x^50 + x^55 + x^66 + x^72 + x^104 + x^120 + x^129 + x^148 + x^183 + x^189 + x^193 + x^212, b(x) = x^25 + x^59 + x^68 + x^128 + x^133 + x^179 + x^186 + x^187 + x^195 + x^197 + x^199 + x^201 + x^203 + x^210 + x^212 + x^215 + x^256; 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^257 - 1 of degree 64, fixing k = 2 deg gcd(a, b, x^m - 1) = 128.", "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. Distance is a witness-backed upper bound from 24 random information-set seeds at 2,000,000 trials each; both witnesses re-verified against these matrices." } }