Covering: $\ell_2(10,2)\le 50$

Write $\ell_2(r,R)$ for the least length of a binary linear code with redundancy $r$ and covering radius $R$. A binary linear code of length 50 and dimension 40 has covering radius exactly 2, so

The November 2025 table (Davydov–Marcugini–Pambianco, arXiv:2511.02542, Table 5.1) had $\ell_2(10,2)\le 51$.

At radius 3, a 33-block partition of the existing 817-column seed and Construction QM$_4^3$ give $\ell_2(41,3)\le26175$, improving Table 7.2's 26238 by 63 columns. The certificate checks all $2^{26}$ seed syndromes and every output column; replay with compute/q16/run_all.sh.

The seed argument

The certificate is a $10\times 50$ parity-check matrix, compute/H_r10_n50.txt: every syndrome in $\mathbb{F}_2^{10}$ is a sum of at most two of its columns (1024/1024, two independent verifiers). The same matrix has a $(2,0)$-partition with $p(H)=10$, so the $\mathrm{QM}_2^2$ construction of arXiv:2511.02542 lifts it to longer codes ($r=18$, $n=815$; $r=20$, $n=1631$) and to the density bound $\bar\mu(2)\le 2601/2048\approx 1.27002$. That is the interesting part: one finite seed moves the asymptotic constant, not just a table entry.

Status

Construction, an upper bound. Sphere covering gives only $\ge 45$, and an $n=49$ search left 7 holes, so 50 is not shown optimal.

Replay

python compute/verify_certificate.py
cd result && ./run_all.sh

Standalone writeup: result/. Explainers: HTML, PDF.

What was tried for 49

Still open. Sphere covering only gives $\ge 45$. Overnight work on 2026-08-21 did not produce a 49, and it did not produce a lower bound.

Three construction families, none of them a neighbour-search on the 50-set:

1. Recover the 2003 Kaikkonen–Rosendahl 51-set and ask whether 51 or 50 is a lift of the $r=8$ record. They are not. Replacing one quotient block of the 50-set by a shorter block does not yield a 49 in any instance that was decided.
2. Prescribe an automorphism of odd prime order. No 49-set is invariant under an element of order 7. Combined with a short arithmetic argument, any 49-set has automorphism group a $\{2,3,5\}$-group. The known 50-set and 51-set are both 2-group symmetric.
3. A fibered construction (one column over every nonzero point of a quotient, plus a kernel block). It reproduces the documented lengths at $r=4,7,8,9$ and is completely decided for $r\le 8$. At $r=10$ it cannot produce any length $\le 47$; its own best is 54. The nearest unfinished case in the family is length 38 at $r=9$, which would beat $\ell_2(9,2)\le 39$ and was not decided.

Code: compute/q9/, compute/q10/, compute/q11/.