Improved upper bounds for partial spreads

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Kurz, Sascha:
Improved upper bounds for partial spreads.
2016
Veranstaltung: Network Coding and Designs , 4.-8.04.2016 , Dubrovnik, Kroatien.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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Abstract

A partial $(k-1)$-spread in $PG(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection such that each point is covered exactly once. So far the maximum size $A_q(n,2k;k)$ of a partial $(k-1)$-spread in $PG(n-1,q)$ was known for the cases $n\equiv 0\pmod k$, $n\equiv 1\pmod k$ and $n\equiv 2\pmod k$ with the
additional requirements $q=2$ and $k=3$. We completely resolve the case $n\equiv 2\pmod k$ for the binary case $q=2$.

Theorem:
For each pair of integers $t\ge 1$ and $k\ge 4$ we have $A_2(k(t+1)+2,2k;k)=\frac{2^{k(t+1)+2}-3\cdot 2^{k}-1}{2^k-1}$.

Theorem:
For integers $t\ge 1$ and $k\ge 4$ we have $A_3(k(t+1)+2,2k;k) \le \frac{3^{k(t+1)+2}-3^2}{3^k-1}-\frac{3^2+1}{2}$.