The existence of (v, 4, 1) disjoint difference families with v a prime power

Abstract

A (v, k, λ) difference family ((v, k, λ)-DF in short) over an abelian group G of order v, is a collection F = {Bi|i ∈ I} of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely λ ways as a difference of two elements lying in some base blocks in F. A (v, k, λ)-DDF is a difference family with disjoint blocks. In this paper, by using Weil’s theorem on character sum estimates, it is proved that there exists a (pn, 4, 1)-DDF, where p = 1 (mod 12) is a prime number and n ≥ 1.