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7, 24, 78, 88, 105, 112, 118, 121, 243, 263, 270, 274, 277, 316, 404, 426, 439, 501 ], [ 8, 25, 79, 89, 106, 113, 119, 122, 244, 264, 271, 275, 278, 317, 405, 427, 440, 502 ], [ 9, 26, 80, 90, 107, 114, 120, 123, 245, 265, 272, 276, 279, 318, 406, 428, 441, 503 ] ] }, "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, of degree 32, which fixes k = 2 deg g = 64 by construction. The pair (a,b) was then drawn from the ideal (g) by information-set decoding over sparse supports, keeping weights 9 and 9. At least one weight is odd, so x+1 does not divide the gcd. 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 = [10, 72, 85, 107, 195, 234, 237, 241, 248] and b = [11, 133, 136, 142, 149, 166, 176, 230, 247].", "notes": "Distance is an upper bound from the lighter of two witnesses, weight 64 on X and 66 on Z, both carried in this file. A 24-seed randomized information-set search at 2,000,000 trials per seed spread 64 to 70 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 70. Folding onto the x^t-invariant subspaces leaves a lightest surviving logical of weight 162. The design point is deg g / m = 0.125." } }