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The divisor g is one such coset's minimal polynomial, of degree 65, which fixes k = 2 deg g = 130 by construction. The pair (a,b) was then drawn from the ideal (g) by information-set decoding over sparse supports, keeping weights 14 and 16; both are odd, so x+1 does not divide the gcd and no weight-m block logical is forced. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 514 qubits at uniform row weight 30, with a = [5, 27, 96, 100, 112, 135, 136, 145, 177, 183, 200, 239, 244, 252] and b = [21, 26, 30, 87, 94, 107, 112, 121, 141, 172, 174, 189, 196, 208, 220, 246].", "notes": "Distance is an upper bound from the lighter of two witnesses, weight 51 on X and 52 on Z, both carried in this file. A 24-seed randomized information-set search at 2,000,000 trials per seed spread 51 to 58 with the minimum found by one seed. A single run of 8,000,000 trials, matching the verifier's own depth, reads 54. A search restricted to logicals supported inside one circulant block, which a general search is blind to, found nothing below 61. Folding onto the x^t-invariant subspaces leaves a lightest surviving logical of weight 122. The design point is deg g / m = 0.253." } }