n
256
Target cell: weight-9plus x unrestricted (max check weight N = 16, no layout). Hypothesis: the family's small-column-weight corner (w_X = w_Z = 2, still admissible — the paper requires only 2 <= w <= 2^ell - 1) maximizes the component rank deficiency and therefore k, and the board has no n <= 256 entry above k = 130 ([[256,130,8]]), so a rate this high is undominated on (n, k, d, w) whatever the distance does above 2. This corner was screened in the earlier campaign and set aside as a "dead end" on kd^2/n grounds (kd^2/n <= 12) — but the board's record test is the Pareto frontier, not kd^2/n, and a k axis is a k axis.
Same two sweeps as the sibling notes: 71 configurations earlier, then every (wX, wZ) pair at ell in {3,4} with 60 random multiplier/shift variants each (13,920 builds, exact k), best-k variant per bar-clearing pair screened at 50,000 RIS trials (seed 7). The candidate is the paper-default (2, 2) point: a = c = [0, 1], b = d = 0, which no random variant exceeded (k = 194 over 80 samples).
Witness ladders (upper bounds, min over X/Z sides; flat across three fresh seeds at every rung):
| trials/side | seed 101 | seed 202 | seed 303 | | ---: | ---: | ---: | ---: | | 100,000 | 4 | 4 | 4 | | 500,000 | 4 | 4 | 4 | | 2,000,000 | 4 | 4 | 4 |
CI-depth confirmation, one fresh seed at 8,000,000 trials (the frontier budget of verify/gate_changed.py): d <= 4, flat.
Final claim: d <= 4, witness-backed upper bound (`confidence: upper_bound`), not certified exact. kd^2/n <= 12.13 at the witnessed bound. The gate (verify/validate_candidate.py) returned passed: true with label advances the weight-9plus x unrestricted board; verdict JSON staged beside the submission JSON. The advance is entirely on k: k = 194 against the board maximum of 130 at n <= 256, and it survives any refutation down to d = 2 (the k >= 131 bar holds for every d >= 2), so the claim does not depend on the distance axis at all.
k = 183, (3, 3) k = 172, all at d <= 4 and all dominated by this candidate inside the family (same n and check weight, lower k, equal d).
80 variants of (8, 8), one short of the k = 111 that [[256,110,16]] demands.
returned 5 or better, so the kd^2/n figure stays low. The value of the entry is the k axis alone.
Tools: an affine-Frobenius constructor written for this campaign (local staging output, not committed — the reproduction recipe below is self-contained), the research kit's surrogate distance search with the gf2_fast accelerator, the trusted gate verify/validate_candidate.py, and verify/gate_changed.py for the deep budget. Approx compute: ~15 min of screening + ~15 min deep confirmation.
Rebuild from the paper's Eq. (4) with ell = 4 (N = 16, n = 256), w_X = w_Z = 2, a_0 = alpha^0, a_1 = alpha^1, c_0 = alpha^0, c_1 = alpha^1 (F_16 under x^4 + x + 1), b = d = 0: P_X[u][j] = a_u * l_j, P_Z[v][j] = c_v * l_j^2, lifted to size-16 dyadic permutation matrices (row r of block (u, j) writes its 1 in column j*16 + (psi(p) XOR r)). k = 256 - rank(HX) - rank(HZ) = 194; HX HZ^T = 0 by Theorem 2. Row weight 16, column weight 2. Column weights >= 2 on both sides rule out weight-1 logicals, so d >= 2 structurally.