Constructions of ( q, k, 1) difference families with q a prime power and k = 4,5

Abstract

For a prime power q ≡ 1 (mod k( k = 1)) does there exist a ( q, k, 1) difference family in GF( q)? The answer to this question is affirmative for k=3 and also for k > 3 provided that q is sufficiently large ( Wilson's asymptotic existence theorem) but very little is known for k > 3 and q not large enough. In this paper we show that for k = 4,5 it is rather easy to find a ( q, k, 1) difference family in a finite field. In particular, by conveniently applying Wilson's lemma on evenly distributed differences, we provide an elementary but very effective method for finding such families. Using this method we succeed in constructing a ( p, 4, 1)-DF for any admissible prime p < 10 6 and a ( q, 5, 1)-DF for any admissible prime power q < 10 4. Finally, we prove that a ( q, 4, 1)-DF exists for any admissible prime power q (which is not prime).