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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^341 - 1 of degree 51, fixing k = 2 deg gcd(a, b, x^m - 1) = 102.", "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. Both single-block kernels have dimension deg gcd(a, b, x^m - 1) = 51 exactly, so the code carries no circulant block wider than k requires. Distance is a witness-backed upper bound from 24 random information-set seeds at 2,000,000 trials each (spread 90-96); an orbit fold over every equal-orbit partition bottoms at 110 and a single-block circulant search at 8 seeds x 600,000 trials finds nothing under 101. Both witnesses re-verified against these matrices." } }