Difference sets and hyperovals (Q1379665)

Let \(q=2^d\), \(2\leq k\leq q-2\) and \((k,q-1)=(k-1,q-1)=1\). The set of points of \(\text{PG} (2,q)\) (represented by homogeneous coordinates) \(D(k)=\{(1,t,t^k)\mid t\in \text{GF} (q)\}\cup\{(0,1,0),(0,0,1)\}\) is a hyperoval iff \(\{t+t^k\mid t\in \text{GF} (q)\setminus\{0\}\}\) is a difference set in the multiplicative group of \(\text{GF}(q)\). So using three infinite families of monomial hyperovals, it is possible to get families of difference sets having the same parameters as Singer difference sets. To begin the study if there are equivalences among the obtained difference sets, known tables are examined.