# Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.

For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ as given by Blokhuis and Mazzocca in , where they also classified the sets attaining this lower bound. For $q$ even the bound is $q(q+1)/2$ and all Kakeya sets attaining this bound are known. Some more examples of "small" Kakeya sets are given in  and .

My question is, what are some possible applications of constructing these small Kakeya sets in $\mathbb{F}_q^2$?

Moreover, what is the state of the art for $n > 2$? What are the best known bounds and the examples that achieve those bounds? Would it be worthwhile to construct explicit examples attaining the bounds there? (which is of course subjective)

 A. Blokhuis and F. Mazzocca. The finite field kakeya problem. In Building Bridges, pages 205–218, 2008. http://link.springer.com/chapter/10.1007%2F978-3-540-85221-6_6

 A. Blokhuis, M. De Boeck, F. Mazzocca, L. Storme. The Kakeya problem: a gap in the spectrum and classification of the smallest examples. Des. Codes Cryptogr., 72 (1) (2014), 21–31.

 J. M. Dover and K. E. Mellinger. Small Kakeya sets in non-prime order planes. European J. of Combin. Volume 47 (2015), pages 95–102.

• Have you seen the papers by Shubhangi Saraf and Swastik Kopparty some with and some without their advisor Madhusudan ? – Anirbit Mar 31 '15 at 3:36
• I am reading one of their papers right now but I don't see any direct application of the explicit examples of small Kakeya sets, as constructed in the papers I have referred to in my question. Perhaps someone with a better understanding of this would be able to shed some light on that. – Anurag Mar 31 '15 at 4:16