On the sunflower bound for k-spaces, pairwise intersecting in a point

Abstract

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space $\mathrm {PG}(n,q)$, where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a code is a sunflower if $|C| > \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) ^2 + \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1$. In this article we will look at the case $t=0$ and we will improve this bound for $q\ge 9$: a set $\mathcal {S}$ of k-spaces in $\mathrm {PG}(n,q), q\ge 9$, pairwise intersecting in a point is a sunflower if $|\mathcal {S}|> \left( \frac{2}{\root 6 \of {q}}+\frac{4}{\root 3 \of {q}}- \frac{5}{\sqrt{q}}\right) \left( \frac{q^{k + 1} - 1}{q - 1}\right) ^2$.