The existence of almost difference families

Almost difference families (ADFs) were introduced by Ding and Yin as a useful generalization of almost difference sets (ADSs), and a number of infinite classes of almost difference families had been constructed. Suppose qq is a prime power. To construct combinatorial designs in GF(q)GF(q), one often needs to find an element x∈GF(q)⧹{0}x∈GF(q)⧹{0}, such that some polynomials in GF(q)[x]GF(q)[x] of degree one or two satisfying certain conditions. Weil's theorem on character sum estimates is very useful to do this. In this paper, a general bound for finding such xx is given. By using this bound and computer searching, some known results on almost difference families by Ding and Yin are improved.