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208, 246, 248, 261, 294, 337, 383, 393, 404, 431, 437, 439, 468, 513 ] ] }, "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^257 - 1 with a(x) = x^0 + x^45 + x^74 + x^76 + x^82 + x^109 + x^120 + x^130 + x^176 + x^219 + x^252, b(x) = x^8 + x^10 + x^48 + x^80 + x^104 + x^126 + x^186 + x^195 + x^200 + x^230 + x^239; 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 48, fixing k = 2 deg gcd(a, b, x^m - 1) = 96.", "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. At m=257, a prime, x^m - 1 has one cyclotomic coset of size 1 and sixteen of size 16; the size-1 coset is the root of x + 1, and admitting it drops the BCH bound on the single-block cyclic code from 74 to 44, so both a and b are drawn with at least one odd weight to keep it out. Both single-block gcds equal 48, matching deg gcd(a, b, x^m - 1). Distance is a witness-backed upper bound from 24 random information-set seeds at 2,000,000 trials each over 10,712 s, with the per-seed spread running 58 to 64. A prime modulus admits no quotient fold, and the orbit fold over every equal-orbit partition of the 514 qubits bottoms at a validated lift weight of 146; a single-block circulant search at 12 seeds x 600,000 trials finds nothing under 58. Both witnesses re-verified against these matrices." } }