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444 ], [ 3, 10, 11, 19, 23, 26, 58, 82, 182, 208, 224, 255, 265, 293, 326, 328, 348, 349, 407, 418, 421, 445 ], [ 4, 11, 12, 20, 24, 27, 59, 83, 183, 209, 223, 225, 256, 266, 294, 327, 329, 349, 350, 408, 419, 422 ] ] }, "family": "generalized-bicycle", "provenance": { "authors": [ "@vprusso" ], "model": "Claude Opus 5", "novelty": "unknown", "origin": "submission", "date": "2026-10-15", "construction": "Cyclic generalized bicycle at m=223, a prime with ord_2(m)=37, so the nontrivial cyclotomic cosets of 2 all have size 37. The divisor g is one such coset's minimal polynomial times x+1, of degree 38, which fixes k = 2 deg g = 76 by construction. The pair (a,b) was then drawn from the ideal (g) by information-set decoding over sparse supports, keeping weights 12 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 = 446 qubits at uniform row weight 22, with a = [23, 26, 37, 95, 96, 116, 118, 151, 179, 189, 220, 222] and b = [13, 39, 139, 163, 195, 198, 202, 210, 211, 218].", "notes": "Distance is an upper bound from a weight-48 logical carried on both sides of this file. The X witness is the one the board's own refutation gate produced against this code's earlier weight-50 claim, at 8,000,000 information-set trials under seed 896171740; the Z witness is its block-swap-and-index-negation mirror, checked against both check matrices and the Z logical basis before being written. A 24-seed randomized information-set search at 2,000,000 trials per seed bottomed at 50, reached by 3 of the 24 seeds, so the spread sits above this claim rather than confirming it and the witness is the binding read. A search restricted to logicals supported inside one circulant block, which a general search is blind to, found nothing below 50 at 12 seeds of 600,000. Folding onto the x^t-invariant subspaces leaves a lightest surviving logical of weight 142 at t=223. The design point is deg g / m = 0.170." } }