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458, 629 ], [ 29, 33, 57, 87, 101, 108, 139, 174, 201, 204, 210, 255, 315, 356, 362, 412, 418, 459 ], [ 30, 34, 58, 88, 102, 109, 140, 175, 202, 205, 211, 256, 316, 357, 363, 413, 419, 460 ], [ 31, 35, 59, 89, 103, 110, 141, 176, 203, 206, 212, 257, 317, 358, 364, 414, 420, 461 ] ] }, "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^315 - 1 with a(x) = x^168 + x^209 + x^215 + x^265 + x^271 + x^312, b(x) = x^57 + x^102 + x^108 + x^111 + x^138 + x^173 + x^204 + x^211 + x^225 + x^255 + x^279 + x^283; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]. The polynomials are low-weight codewords of the cyclic code generated by a degree-25 divisor of x^315 - 1, which fixes k = 2 deg gcd(a, b, x^m - 1) = 50.", "notes": "Designed-divisor construction: a and b are drawn as low-weight codewords of the ideal of a chosen divisor of x^m - 1, so the dimension is selected rather than inherited. The X witness was found by a symmetric fold onto x^t-invariant vectors and is not reachable by random information-set decoding, which bottoms out at weight 90 over 24 seeds at 2,000,000 trials each; the Z witness is from that scan. Both were re-verified against these matrices." } }