Orthomorphisms and the construction of projective planes

We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field $GF(9)$ as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four self-dual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the “sporadic” (not translation) plane of Mathon.