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10, 31, 50, 60, 107, 191, 238, 248, 257, 282, 306, 321, 374, 389, 408, 410, 432, 438, 457, 472 ] ] }, "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^41 + x^56 + x^75 + x^81 + x^103 + x^105 + x^124 + x^139 + x^192 + x^207 + x^231 + x^256, b(x) = x^8 + x^18 + x^65 + x^149 + x^196 + x^206 + x^225 + x^246; 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 49, fixing k = 2 deg gcd(a, b, x^m - 1) = 98.", "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. The 24-seed general search at 2,000,000 trials each bottomed at 58, but a logical supported on a single circulant block is constrained only by that block, and here that kernel has dimension deg gcd(b, x^m - 1) = 65 against the 49 that k requires. A single-block search at 8 seeds and 600,000 trials finds weight 54 there, so the claim is 54 on X with 58 still standing on Z. Both witnesses re-verified against these matrices." } }