Covering radius 3 at redundancy 44

Result

The run certified

improving the 52415 upper bound in Davydov–Marcugini–Pambianco, arXiv:2511.02542v1, Table 7.2, by 64 columns.

Construction

The notebook's existing $26$ by $817$ radius-3 matrix is a QM₅³ construction. Its proof supplies a 34-block $(3,0)$-partition: 23 Golay $A$-blocks, 10 blocks from the embedded 50-column radius-2 matrix, and one W₅ block. Refining that partition to 65 blocks supplies all indicators in $\mathbb F_{64}\cup\{*\}$ for QM₄³ with $m=6$. The resulting parameters are

Certificate

The Python builder reconstructs the 817-column parent before emitting the partitions and output matrix. An independent C verifier:

• exhausts all 67,108,864 seed syndromes under both the 34- and 65-block partitions;
• checks that the latter is a refinement;
• checks $\mathbb F_2[x]/(x^6+x+1)$ and all 65 indicators;
• checks ranks, distinct nonzero columns, and all 52,351 QM₄³ column identities.

The syndrome bitmap is 8 MiB. The certificate uses Theorem 6.1 for the lift and does not enumerate $2^{44}$ syndromes.

Replay from problems/covering/:

compute/q13/run_all.sh