On extendable planes, M.D.S. codes and hyperovals in PG(2, q), q=2^t

Abstract

The question of the existence of finite planes seems to be beyond the range of present techniques. In this paper we skirmish with an easier question: extendable planes. We show how extendable planes arise as special cases of certain maximum distance separable codes (M.D.S. codes). A synthetic characterization of extendable planes is obtained. A different characterization is obtained in terms of hyperoval systems. Moreover, since π=PG(2, q), q=2^t, is extendable this leads to new insights concerning the subtle and marvellous structure of certain hyperoval systems in π. A priori, it seems somewhat surprising that very much can be said about hyperovals in π, as they have certainly not been classified. In particular, we obtain a partial generalization of the famous ‘even intersection’ property of hyperovals in PG(2, 4). We conclude with a discussion of hyperoval ‘spreads’ and ‘packings’ in π along with some open questions.