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In design theory the following is the defintion of a packing :

Definition : A $(v,k)$-packing is a pair $(V, \mathcal{B})$ of a finite set $V$ of cardinality $\vert V \vert = v$ and a finite set $\mathcal{B}$ of $k$-subsets of $V$ is a of order $v$ and block size $k$ such that any pair of elements of $V$ appears in at most one element of $\mathcal{B}$. The elements of $S$ are the points, and those of $\mathcal{B}$ are the blocks.

One of the interesting problems related to packings is finding a $(v,k)$-packing with the maximal number of blocks. Let $D(v,k)$ denote the number of blocks in a maximal packing. This is a well-studied problem. I am interested only in a particular case $k=6$ but I could not find any papers studying it?

Question: What is the minimum number $v_0$ for which $D(v_0,6)\geq 100$ ? is there any online database for packings (like the one available for coverings)?

It know that $66 \geq v_0\geq 56$, the first bound is because of the existence of a block design $(66,6,1)$ and the second bound follows from the Johnson bound $$D(v,k) \leq \left\lfloor \frac v k \left\lfloor \frac {v-1}{k-1}\right\rfloor \right\rfloor$$.

Edit: if I understand well the relation between packings and constant weight codes then what I am looking for is the smallest $v_0$ such that $A(v_0,10,6)\geq 100$ in this website there is a lot of bounds on those numbers, in particular, $A(60,10,6) \geq 104$ from $TD(6,10)–TD(6,2)$ (with $96$ blocks) and adding six $6$-lines in the groups and two $6$-lines in the hole. But I don't know how to construct the packing from this information?

This implies that $v_0 \leq 60$ (only four values remaining, probably an open problem)

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    $\begingroup$ There should be some tight relation between packing and coverings. Something like a packing for n, k leads by complementation to a covering for n,n-k. Check out Handbook of Combinatorial Designs. Also check out La Jolla repository. Gerhard "Little Fuzzy On Packing Thinking" Paseman, 2020.01.01. $\endgroup$
    – Gerhard Paseman
    Commented Jan 2, 2020 at 0:05
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    $\begingroup$ @Gerhard, it took me time, but I did not manage to find your article on la Jolla Repository, do you have a link ? Thank you ! $\endgroup$
    – Elaqqad
    Commented Jan 2, 2020 at 1:07
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    $\begingroup$ The website has been rearranged. The older version had a link to a paper by many authors (I believe Greg Kuperberg was one), and my sometimes faulty memory tells me that packing was related to covering in that paper (this may be a wrong assertion). At this point, you should wait awhile until someone with a better memory responds here, and then contact Dan Gordon if you don't get satisfaction here. Gerhard "Sorry To Raise Hope Prematurely" Paseman, 2020.01.01. $\endgroup$
    – Gerhard Paseman
    Commented Jan 2, 2020 at 1:22
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    $\begingroup$ The CRC Handbook of Combinatorial Designs, 2nd ed., (2007) has seven pages on packings. Skimming through those pages (email me for a copy) says almost nothing about $k=6$ and I know a lot of work went into $k=5$, so I suspect your question is pretty open? The CRC section does reference an old survey by W. H. Mills and R. C. Mullin, "Coverings and Packings" in Contemporary Design Theory, Wiley, 1992. $\endgroup$
    – Ken W. Smith
    Commented Jan 2, 2020 at 3:49
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    $\begingroup$ Perhaps, Gerhard means this article arxiv.org/abs/math/9502238 $\endgroup$
    – Max Alekseyev
    Commented Jan 2, 2020 at 12:48

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