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 q is a prime power. To construct combinatorial designs in GF ( q ) , one often needs to find an element x ∈ GF ( q ) ⧹ { 0 } , such that some polynomials in GF ( q ) [ x ] of degree one or two satisfying certain conditions. Weilʹs theorem on character sum estimates is very useful to do this. In this paper, a general bound for finding such x is given. By using this bound and computer searching, some known results on almost difference families by Ding and Yin are improved.