Covering radius 3 at redundancy 41

On 2026-09-23, q16 found a 33-block $(3,0)$-partition of the notebook's $26\times817$ radius-3 seed by merging inherited blocks 7 and 31. An exact search of all 561 single merges found two valid pairs. Construction QM$_4^3$ with $m=5$ gives

$\ell_2(41,3)\le26175,$

improving the published 26238 and the notebook's previous 26206. The independent certificate sweeps all $2^{26}$ seed syndromes and verifies every output-column identity. It uses the construction theorem for output coverage, without a $2^{41}$ syndrome sweep. Replay from problems/covering/ with sh compute/q16/run_all.sh.

An exact representation count also shows that every one of the 817 seed columns is essential for radius 3, even without partition restrictions. This rules out a one-column deletion of this seed. The $r=24$ alternative-seed and $r=10$ two-block handles remain open in compute/LEAVES.md.

Published comparison: Davydov–Marcugini–Pambianco, Table 7.2.