Notes from authors — 2026-08-19

Do not add the source PDF to the git repo. Citation only; no figures, no long excerpts.

Same paper as arXiv:2511.02542. IEEE author’s accepted version of Davydov–Marcugini–Pambianco, New upper bounds for binary linear covering codes, IEEE Trans. Inform. Theory (accepted 2026), DOI 10.1109/TIT.2026.3716379 = arXiv:2511.02542 (4 Nov 2025). Same title, abstract bounds, Table 5.1 (still $\ell_2(10,2)\le 51$), and QM numbering. Not a later paper. Extract unused tactics only. Already used from this paper: QM$_2^2$ (Thm 4.1 / (4.2)), QM$_3^2$ (Thm 5.1), QM$_5^2$ (Thm 5.4(ii)), 10-block $(2,0)$-partition of the 50-set, theorem-only QM$_2^2$ at $r=42,44$.

Bibliographic facts (from the PDF)

• Authors: Alexander A. Davydov (IITP RAS); Stefano Marcugini, Fernanda Pambianco (Perugia).
• MSC 94B65, 94B25, 94B60, 94B05.
• Correspondence they use: $\ell_q(r,R)=s_q(r-1,R-1)$ (1-saturating sets in $PG(r-1,q)$).
• Computer-search method is not specified (Thms 5.2, 7.1, Lemma 7.5 just say “computer search”). Magma is cited [9]; no anneal/SAT recipe.

Unused constructions (R = 2)

QM$_{2}^{1}$ (Thm 4.1 / (4.1), (4.3); from [27]). $B\subseteq\mathbb F_{2^m}\cup\{*\}$, $2^m+1\ge p(H_0)$, $D=D_2(2)$ (two Hamming blocks). Length $n=2^m(n_0+2)-2$ is worse than QM$_2^2$, but $p(H_C)\le p(H_0)$ — the partition size is preserved, not grown to $2^{m+1}+1$. Use as an intermediate when a later lift is blocked by a large inherited $p$.

Block-splitting to hit exactly $2^m$ (Thm 5.4, QM$_{2}^{4}$, display (5.9)). They take the computer-searched 11-block 2-partition $P_{KR}$ of the Kaikkonen–Rosendahl 51-set (Thm 5.2 / (5.3)) and split the first three blocks so the dependent triple $\{h_5,h_{27},h_{29}\}$ becomes three singletons and $p=16=2^4$. Same indicator per block, $B=\mathbb F_{2^4}$, $D=D_1(2)$. We already isolate $(491,734,821)$ for QM$_5^2$; the unused move is splitting an existing good partition until $p=2^m$ (or $2^m+1$ for QM$_{2}^{1}$) to unlock a smaller $m$.

Explicit lift-partition, not the inequality (proof of Thm 5.4). Constructive 2-partition of $H_C$: (i) interim blocks = seed blocks, copied across each $A(h_j,\beta_j)$; (ii) split every interim block into $\{\xi=0\}$ vs the rest; (iii) $D$ is its own last block. QM$_{2}^{2}$ gives $p\le 2^{m+1}+1$. QM$_{2}^{5}$ with a $*$ block: do not split the $*$ block, so $p\le 2^{m+1}+2$. This is how to compute $p(H_C)$ instead of inheriting the paper bound — the stated soft spot.

QM$_{2}^{6}$ (Thm 5.6) is just iterated QM$_{2}^{2}$ on $\Phi$-length seeds with the window $t_0\ge m\ge\lambda_0+1$. Iteration recipe is in the proof of Thm 5.7; $p$-tracking for even $r\ge 42$ is (5.16).

Unused constructions (other radii) — plug the 50-set in

QM$_{3}^{5}$ (Thm 7.3, new here). Start from a radius-3 code $C_0$ (they use the perfect Golay $[23,12,7]_2 3$, trivial $(3,0)$-partition, $p=23$) and glue a radius-2 code $V_{2m}$ as $D=D_4$. $n=2^m(n_0+1)+n_{2m}-1$, $r=r_0+3m$. They plug KR / $\Phi$ as $V_{2m}$. Our shorter $V_{2m}$ beat their Table 7.1/7.2 on the spot:

Need the Lemma 7.5 analogue: every vector of $\mathbb F_2^{10}$ is a 3-sum of the 50-set (they verified this for KR by computer search; it gives $\ell_C=2$ at $r=26$). Remark 7.4 (proof omitted): if $\ell_C=2$ and $m\ge 3$, they claim $p(H_C,2)=5p(H_0,0)+n_{2m}+3$.

QM$_{3}^{4}$ from the Östergård–Kaikkonen $[18,9]_2 3$ seed (Thms 6.4, 7.1–7.2). Explicit 11-block $(3,1)$-partition $P_{OK}$; $m=4$ gives $\ell_2(21,3)\le 303$. Hex last entry is printed ICE — treat as OCR until checked against [65].

QM$_{4}^{4}$ (Thm 9.1, new here). Odd $m$, $D=D_5$ contains a radius-2 $H_{2m}$. They use OK2 $[19,8]_2 4$ + KR to get $\ell_2(31,4)\le 690$. Our 50-set: $n=2^5(19+1)+50-1=689$.

Unused $D$-blocks vs what we built: $D_2(2)$ (QM$_{2}^{1}$), $D_3$ (QM$_{3}^{3/4}$), $D_4$ (QM$_{3}^{5}$), $D_5$ (QM$_{4}^{4}$). We already use $D_1(2)$ and $D_6$.

Table 5.1 still weak (their “best as far as we know”)

• All odd $r$ still the old $f(r)=5\cdot 2^{(r-3)/2}-1$ family (Thm 4.2(ii) / [32]). Densities $\approx 1.52$–$1.56$. Open problem §10: new families for $R=2$, $r=2t-1$. First holes: $r=9$ ($n=39$), $r=11$ (79), $r=13$ (159). ($r=8,9$ already searched; no dent.)
• Even leftovers not in $\Phi$: $r=12$ (107), $14$ (215), $16$ (431) — old $\varphi$ / [27]. No QM from the 50-set hits these $r$.
• They never improve $n<51$ at $r=10$. Open problem §10: improve sporadic codes that are not in an infinite family — that is $n=49$.

How to get below 49

The paper has no shortening construction at $r=10$. Their computer search is for partitions of a fixed $H$, not for a shorter $H$. $\ell_2(10,2)=s_2(9,1)$: a 49-column code is a 1-saturating 49-set in $PG(9,2)$. Deletion from the 50-set leaves $\ge 9$ holes — need a different 49-set, not a subset. Possible starts: KR 51 minus two with a different repair; geometric 1-saturating-set search; do not expect a QM lift (lifts increase $r$).

How to tighten $p(H)$

1. Implement the Thm 5.4 constructive lift-partition on our $r=18,20,28$ matrices and count $p$, then computer-search a smaller one.
2. Concrete unlock: our $r=28$, $n=26111$ has $p\le 66$, so QM$_2^2$ needs $2^m\ge 66$ ($m\ge 7$). If a search gets $p\le 64$, $m=6$ gives $\ell_2(40,2)\le 64\cdot 26111+63=1671167$ (table: 1703935). $p\le 33$ would unlock $m=5$ and $r=38$.
3. QM$_{2}^{1}$ if we would rather keep $p$ than shorten $n$.
4. For QM$_{3}^{5}$: a dependent triple in three distinct blocks makes $p(H_C,1)=p(H_C,0)$ (saves 2).

Other radii / open list (§10)

They flag: $R=2$ odd $r$; $R=3$ residues $3t-2$ and $3t$ (the $\varphi,\psi$ families, densities $\approx 2.25$ and $1.90$); $R=4$ residues $4t-1,4t-2,4t-3$; and no infinite families for $R\ge 5$. Methods “can be used” for $R\ge 5$; only sporadic examples exist.

Check before citing (do not trust the PDF blindly)

• KR hex (4.9) and $P_{KR}$ (5.3) / $P_{KR}^*$ (5.9) against [50].
• $h_5+h_{27}+h_{29}=0$ (Thm 5.2(ii)).
• Lemma 7.5 on KR, and the same 3-sum property on our 50-set.
• $M_{OK}$ last word ICE (6.10) — invalid hex; independently recovered as 1CE (unique among 512 last words). See compute/recover_mok.py.
• Table 5.1 commas (1,56219) are decimal-comma typos.
• Abstract densities vs (5.15): $\mu(2)\approx 262/2^9\approx 1.32031$.
• IEEE vs arXiv v1: abstract numbers match; we did not line-diff the body.