points with projective coordinates (1,0,0), (0,1,0), (0,0,1) and (1,1,1) are contained in any hyperoval. The remaining points of the hyperoval (when h > 1) will have the form (t, f(t),1) where t ranges through the values of the finite field GF(2h) and f is a function on that field which represents a permutation and can be uniquely expressed as a polynomial of degree at most 2h - 2, i.e. it is a permutation polynomial. Notice that f(0) = 0 and f(1) = 1 are forced by the assumption concerning the inclusion of the specified points. Other restrictions on f are forced by the no three points collinear condition. An f which produces a hyperoval in this way is called an o-polynomial. The following table lists all the known hyperovals (as of 2011) of PG(2,2h) by giving the o-polynomial and any restrictions on the value of h that are necessary for the displayed function to be an o-polynomial. Note that all exponents are to be taken mod(2h - 1). Known Hyperovals in PG(2,2h) a) The Subiaco o-polynomial is given by: whenever , where tr is the absolute trace function of GF(2h). This o-polynomial gives rise to a unique hyperoval if and to two inequivalent hyperovals if . b) To describe the Adelaide hyperovals, we will start in a slightly more general setting. Let F = GF(q) and K = GF(q2). Let be an element of norm 1, different from 1, i.e. bq+1 = 1, . Consider the polynomial, for , f(t) = (tr(b))−1tr(bm)(t + 1) + (tr(b))−1tr((bt + bq)m)(t + tr(b)t½+ 1)1−m + t½, where tr(x) = trK/F(x) = x + xq. When q = 2h, with h even and m = ±(q - 1)/3, the above f(t) is an o-polynomial for the Adelaide hyperoval. c) The Penttila-O'Keefe o-polynomial is given by: f(t) = t4 + t16 + t28 + η11(t6 + t10 + t14 + t18 + t22 + t26) + η20(t8 + t20) + η6(t12 + t24), where η is a primitive root of GF(32) satisfying η5 = η2 + 1. Hyperovals in PG(2, q), q even, q ≤ 64 As the hyperovals in the Desarguesian planes of orders 2, 4 and 8 are all hyperconics we shall only examine the planes of orders 16, 32 and 64. PG(2,16) In the details of a computer search for complete arcs in small order planes carried out at the suggestion of