Translation nets and fixed-point-free group automorphisms

Certain translation nets are shown to be equivalent to sets of mutually orthogonal Latin squares constructed by the automorphism method. Known results on fixed-point-free automorphisms are used to improve the known upper bounds on the maximum number of parallel classes in such a net. In particular, the maximum number is found exactly for such nets whose translation groups are Abelian. Applications are given both to the statistical design of experiments and to other parts of pure mathematics.