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154, 186, 206, 212, 213, 219, 230, 288, 349, 362, 383, 433, 442, 461, 464 ], [ 1, 133, 134, 155, 187, 207, 213, 214, 220, 231, 289, 350, 363, 384, 434, 443, 462, 465 ], [ 2, 134, 135, 156, 188, 208, 214, 215, 221, 232, 290, 351, 364, 385, 435, 444, 463, 466 ] ] }, "family": "generalized-bicycle", "provenance": { "authors": [ "@vprusso" ], "model": "Claude Opus 5", "novelty": "unknown", "origin": "submission", "date": "2026-10-15", "construction": "Cyclic generalized bicycle at m=257, a prime with ord_2(m)=16, so the nontrivial cyclotomic cosets of 2 all have size 16. The divisor g is a product of the minimal polynomials of 2 cyclotomic cosets of size 16, times x+1, of degree 33, which fixes k = 2 deg g = 66 by construction. The pair (a,b) was then drawn from the ideal (g) by information-set decoding over sparse supports, keeping weights 8 and 10. Both weights are even, so x+1 divides the gcd; m is prime, so the only block logical this forces has weight m and is far above the claim. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 514 qubits at uniform row weight 18, with a = [47, 50, 69, 78, 128, 149, 162, 223] and b = [24, 35, 41, 42, 48, 68, 100, 121, 122, 254].", "notes": "Distance is an upper bound from the lighter of two witnesses, weight 63 on X and 63 on Z, both carried in this file. A 24-seed randomized information-set search at 2,000,000 trials per seed spread 63 to 68 with the minimum found by one seed. A single run of 8,000,000 trials, matching the verifier's own depth, reads 66. A search restricted to logicals supported inside one circulant block, which a general search is blind to, found nothing below 73. Folding onto the x^t-invariant subspaces leaves a lightest surviving logical of weight 178. The design point is deg g / m = 0.128." } }