n
82
The board carries 169 generalized-bicycle entries and, at the time of this submission, none of them at check weight 4. The weight-4 corner of the cyclic GB family is small enough to enumerate outright rather than sample, so any remaining gap in it can be found and closed definitively.
A weight-4 cyclic GB code is
H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]
over the ring x^m - 1, with a and b each of weight 2. A monomial factor on either polynomial is a cyclic shift of the corresponding block, so a and b can be normalised to a = 1 + x^i and b = 1 + x^j. Two further symmetries act on the pair: the unit map x -> x^u for u coprime to m, applied to both polynomials at once, and the a/b swap, which exchanges the two qubit blocks. Quotienting by all three leaves roughly m^2 / (2 phi(m)) orbits per ring, which is small: m = 499 has 250 orbits rather than 124,251.
k = 2 deg gcd(a, b, x^m - 1) = 2 gcd(i, j, m). This is not a free parameter. Within a block, qubit c is linked to c +- i through an H_X row and to c +- j through an H_Z row, so block connectivity is generated by the subgroup <i, j> = <gcd(i,j,m)> and the Tanner graph splits into exactly gcd(i,j,m) components. A code with k > 2 in this family is therefore a direct sum of gcd(i,j,m) smaller copies, and fails the verifier's tanner_connected and stabilizer_group_connected checks. [[80,10,4]] from the same sweep is five disjoint copies of [[16,2,4]].
The connected part of the family is exactly gcd(i,j,m) = 1, which forces k = 2. Six of the eight candidates the first pass produced died on this, which is why the sweep now tests it before spending any distance work on an orbit.
Rings m = 40 upward, every orbit, with k computed from gcd before any distance search. An orbit is screened only if some d makes (n, k, d, 4) undominated on the live board; the threshold comes from a binary search on the board itself, so orbits that cannot land anywhere are skipped without a search.
This code is m = 41, a = 1 + x, b = 1 + x^9.
Random information-set search on each side separately, 200k then 800k then 800k trials, driving the X and Z sides with independent calls so that a run which happens to find one side cannot leave the other without a witness. Ladder: 9, 9, 9 on both sides. Both witnesses re-checked against H directly (in the kernel of the opposite matrix, outside the row space of its own).
The verifier's own refutation pass found nothing lighter in 5,780 RIS trials at seed 256858330.
[[82,2,9]] against the board's [[84,2,9]] at the same k, d and weight: two fewer physical qubits. d / sqrt(n/2) = 1.406, against a ceiling of sqrt(2) for this family, so there is very little room left on this curve. The value of the sweep is that it is exhaustive: when it finishes, the remaining gaps in the weight-4 k=2 curve are known rather than estimated.
m = 41, a = 1 + x, b = 1 + x^9 H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]
The distance is an upper bound from witness search, not a proof. Literature novelty is unverified: weight-4 cyclic GB codes are a classical family and these parameters may well appear in the 2BGA literature.