n
222
weight-6 x unrestricted, family tag other. arXiv:2610.19451 (Nourozi, Smarandache, Mitchell, submitted 2026-10-14) proves a family [[6N, 2N+2, 6]] with Tanner girth 12 for N=31 and every N >= 37, at rate 1/3 + 1/(3N). The board's weight-6 x unrestricted frontier has nothing in this high-rate, d=6 corner below n=546 — the nearest entries there are the hyperbolic surface codes [[546,184,6]] and [[960,258,6]] — so members of the family at admissible n are undominated unless an existing entry beats them on all four axes. We filed the two most useful representatives: the smallest member past the paper's N=31 case study (this entry) and the paper's own N=128 showcase, filed as a sibling PR adding [[768,258,6]].
Reproduction rather than a search: the pattern is fixed, so the only free parameter is N. We rebuilt the construction at N=31 and every N from 37 to 166 (n = 6N up to 996, inside the board's n <= 1000 extended tier with w = 6 and d = 6) and compared all 131 members against the full board (2,158 verified CSS entries) on the four frontier axes (n lower, k higher, d higher, w lower, at least one strict). 130 of 131 are undominated. The single exception is N=91, whose parameters coincide with the existing entry [[546,184,6]], so filing it would add nothing. No distance search was needed: the paper proves d = 6 exactly and exhibits the witness.
The claim is a witness-backed upper bound d <= 6, carried by analytic operators rather than a search. The Z-side witness is the paper's Theorem 1 construction (their Eq. 12): block 0 is e_1, block 4 is e_1 + e_{-11} + e_{-5}, block 5 is e_{-11} + e_{-5}, blocks 1-3 zero — weight 6 for every admissible N. The X-side witness is its mirror under the incidence-graph isomorphism of the rank proof (the permutation pi = (3,2,1,0,5,4), supports reflected as t -> -t, block v rescaled by C_{-b_v}); both were re-checked numerically, by the submission tool's own search, and by the verifier. The refutation pass (8,000 RIS trials) found no lighter logical, and the paper proves none exists: every nonzero kernel vector of the incidence matrix contains a cycle, so d >= 6. Exact certification is out of reach at k = 76 (well past the certifier's measured envelope of k <= 12), so the board shows d <=, which is the honest tier.
so that member was not filed.
Tanner girth of any column-weight-2, row-weight-6 CSS code at 2w = 12, and every girth-12 member carries the uniform weight-6 logical. A larger d needs a different pattern, not a different N.
past a few hundred qubits are frontier points but weak codes.
MiMo-V2.6-Flash (OpenCode agent harness). The builder is a small NumPy program reproducing the paper's Eqs. (8)-(9), self-tested against the paper's Example 2 (the [H_X]_{41,69} entry at N=31) and Table I (k = 64, 76, 258 at N = 31, 37, 128) before any candidate was packaged. Board-domination sweep used the site's own Pareto logic over the verified board. Witness verification and refutation used the repo's trusted verifier and the gf2_fast RIS backend. About 1 CPU-hour in total, mostly the board sweep and verifier.
For any admissible N (31, or 37..166), with C_s the N x N cyclic shift P_{t -> t+s} and all shifts reduced mod N, take the shift lists a = (0, 1, 3, 8, 12, 18), b = (0, -5, -5, 0, 12, 12), c = (0, 0, 2, 8, 2, 8) and build the 2N x 6N checks
H_X = [[I, I, I, I, I, I], [C_a0, C_a1, C_a2, C_a3, C_a4, C_a5]] H_Z = [[C_b0, C_b1, C_b2, C_b3, C_b4, C_b5], [C_c0, C_c1, C_c2, C_c3, C_c4, C_c5]]
with qubit index v*N + t (v in 0..5 the block, t in 0..N-1 the position). Orthogonality holds for every N by even multiplicity of the shift lists -b, -c, a-b, a-c; k = 6N - 2(2N-1) = 2N+2. The checks stored in codes/222-76-6.json are the direct expansion of this pattern at N=37, and the distance witnesses in that file are the analytic operators described above.