For a prime power q = 1 (mod k(k-1)) does there exist a (q, k, 11 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 theoremt 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< l0 6 and a (q, 5, 1)-DF for any admissible prime power q < 104. Finally, we prove that a (q, 4, 1 )-DF exists for any admissible prime power q (which is not primej.
Recall the definition of a simple difference family.
Definition 1. Let o~ = {B1 ..... Bt} be a family of k-subsets of an additive group G of order v. We say that ~ is a (v, k, 1) simple difference family (briefly (v, k, 1)-DF) if any nonzero element of G can be represented exactly in one way as a difference of two elements lying in a same member of ,f.
The members of a DF are called base blocks. A (v,k, I)-DF in G is called cyclic, briefly (v, k, 1)-CDF, when G is isomorphic with 7/~.
We obviously have v-1 (rood k(k-1)) for any (v, k, 1)-DF.
We also recall that any simple difference family generates a Steiner 2-design. In particular a (v, k, 1)-CDF gives rise to a (v, k, 1) Steiner 2-design with an automorphism consisting of a single cycle of length v, i.e. a cyclic Steiner 2-design. As in , we denote such a design by CS(2, k, v).
In this paper we are interested in simple difference families in finite fields. We use the following notation: for q a prime power, H denotes the multiplicative group of the Elsevier Science B.V.