The small Kakeya sets in $T^{*}_{2}(\mathcal{C})$, $\mathcal{C}$ a conic

Abstract: A Kakeya set in the linear representation $T<sup>{*}_{2}(\mathcal{C})$, $\mathcal{C}$ a non-singular conic, is the point set covered by a set of $q+1$ lines, one through each point of $\mathcal{C}$. In this article we classify the small Kakeya sets in $T<sup>{*}_{2}(\mathcal{C})$. The smallest Kakeya sets have size $\left\lfloor\frac{3q<sup>{2}+2q}{4}\right\rfloor$, and all Kakeya sets with weight less than $$\left\lfloor\frac{3(q<sup>{2}-1)}{4}\right\rfloor+q$$ are classified: there are approximately $\sqrt{\frac{q}{2}}$ types.