The binary code arising from a 2-design with a nice collection of ovals

In the Fall of 1976 Jack van Lint and I undertook an investigation of ovals in projective designs. Later a student of m ine, Bruno Andriamanal imanana, pushed these ideas somewhat further. Although the theorem I wish to discuss is implicit in these works, a concise statement has not appeared. I would like, therefore, to take the opportunity of honor ing Jessie MacWilliams by presenting a theorem I am quite certain she will like. The theorem has some interesting corollaries; they are, as far as I know, entirely new. One is a result concerning the possible projective plane of order 10. Roughly speaking, it says that if a plane of order 10 has a 2-design of ovals, then it is very likely extendable to a 3-design on 112 points.