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141, 184, 186, 192, 269, 301, 302, 327, 335, 385, 429, 442, 484 ], [ 18, 20, 24, 51, 71, 113, 123, 142, 185, 187, 193, 270, 302, 303, 328, 336, 386, 430, 443, 485 ] ] }, "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^28 + x^70 + x^83 + x^127 + x^177 + x^185 + x^210 + x^211 + x^243, b(x) = x^63 + x^69 + x^71 + x^114 + x^133 + x^143 + x^185 + x^205 + x^232 + x^236 + x^238; 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 32, fixing k = 2 deg gcd(a, b, x^m - 1) = 64.", "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. m = 257 is prime, so there is no quotient fold at all; the orbit fold over every equal-orbit partition of the 514 qubits returns a validated lift of weight 162, and a single-block circulant search at 12 seeds x 600,000 trials finds nothing under 76. Both single-block gcds equal 32, matching deg gcd(a, b, x^m - 1). Draws with both polynomial weights even are rejected, since (x+1) then divides the gcd and adds the root-1 coset, which drops the single-block BCH bound from 86 to 52 at this degree. Distance is a witness-backed upper bound from 24 random information-set seeds at 2,000,000 trials each over 28,956 s, with the per-seed spread running 64 to 70; one seed of the 24 found 64 and the rest sat at 66 or above. Both witnesses re-verified against these matrices." } }