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 is 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 $k\leq7$. As a consequence, we get new difference families and hence new Steiner 2-designs.