Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a ( q , 5, 1) difference family in (GF( q ), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a ( q , 4, 1) difference family in (GF( q , +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2‐designs with block sizes 4 and 5. In particular, we get a ( p 4α , 5, 1)‐DF for any odd prime p ≡ 2, 3 (mod 5), and a ( p 2α , 4, 1)‐DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.