Small Kakeya sets in non-prime order planes

The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in AG(2,s)AG(2,s), ss a prime power, the smallest possible size of such a set is 12s(s+1). In this paper we examine a new construction of an infinite family of sets in AG(2,s)AG(2,s) containing a line in every direction, where s=qes=qe with qq a prime power and ee an integer with e>1e>1. These sets have size 12q2e+O(q2e−1), which is small in the sense that the lower bound is also 12q2e plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction.