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

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 6 and a ( q , 5, 1)-DF for any admissible prime power q 4 . Finally, we prove that a ( q , 4, 1)-DF exists for any admissible prime power q (which is not prime).

Marco Buratti | M. Buratti

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