α-Flocks with Oval Herds and Monomial Hyperovals

InPG(3,q),qeven, Cherowitzo made a detailed study of flocks of a cone with a translation oval as base; also calledα-flocks [4]. To a flock of a quadratic cone inPG(3,q),qeven, there always corresponds a set ofq+1 ovals inPG(2,q), called anoval herd. To anα-flock of a cone with an arbitrary translation oval as base, there corresponds a herd ofq+1 permutation polynomials. For some, but not for all, known examples ofα-flocks, theseq+1 permutation polynomials define an oval herd. This leads to the fundamental problem of determining whichα-flocks correspond to an oval herd. This article studies a class ofα-flocks and explicitly describes which members of this class have an associated oval herd. To achieve this goal, all monomial hyperovals {(1,t,tk) t Fq} {(0, 1, 0), (0, 0, 1)} withk=2i+2j,i≠j, are determined.