Certified radius-3 bounds

Status: certified dent (2026-09-23).

The exact target inequalities are

Davydov–Marcugini–Pambianco, arXiv:2511.02542v1, Table 7.2, lists 26238 and 104831, respectively. The improvements are 63 and 128 columns. This notebook previously had $\ell_2(41,3)\le26206$; the first inequality improves that by 31 columns.

The starting 26 by 817 matrix is compute/H_R3_r26_n817.txt. Its 34-block $(3,0)$-partition is inherited from QM$_5^3$. Merging blocks 15 and 24 preserves coverage of all $2^{26}$ syndromes and leaves 33 blocks. Applying Construction QM$_4^3$ (Theorem 6.1, (6.4), (6.8)) with $\mathbb F_{32}=\mathbb F_2[x]/(x^5+x^2+1)$ gives

For the second bound, a 129-block refinement of the inherited partition provides all indicators in $\mathbb F_{128}\cup\{*\}$, with $\mathbb F_{128}=\mathbb F_2[x]/(x^7+x+1)$. The same construction gives

Run compute/q15/run_all.sh from problems/covering/. Its independent C checkers exhaust the seed syndrome space under the 33- and 129-block partitions, verify full rank and distinct nonzero columns, test both fields, and compare every generated output column with the QM$_4^3$ identity. The output syndromes are covered by Theorem 6.1; no $2^{41}$ or $2^{47}$ sweep is claimed. The sphere-covering volume at radius two is below $2^r$ in both cases, so the constructed codes have radius exactly three.

Falsifiers: a missing seed syndrome under either partition, a duplicate or zero output column, rank below 41 or 47, a wrong field or indicator, an incorrect output column, or a counterexample to the cited construction theorem. No optimality claim.