On simple radical difference families

For q a prime power and k odd (even), we define a ( q,k ,1) difference family to be radical if each base block is a coset of the k th roots of unity in the multiplicative group of GF( q ) (the union of a coset of the ( k − 1)th roots of unity in the multiplicative group of GF( q ) with zero). Such a family will be denoted by RDF. The main result on this subject is a theorem dated 1972 by R.M. Wilson; it is a sufficient condition for the existence of a ( q,k , 1)‐RDF for any k . We improve this result by replacing Wilson's condition with another sufficient but weaker condition, which is proved to be necessary at least for k ⩽ 7. As a consequence, we get new difference families and hence new Steiner 2‐designs. © 1995 John Wiley & Sons, Inc.