Let (G,+) be a group of order v, let D be a k-subset of G, and for each w in G define d_D(w) to be the cardinality of the intersection of the sets D+w and D. Then D is said to be a (v,k,λ,t) almost difference set in G if d_D(w) takes on the value λ, t times, and the value λ+1, v-t-1 times, when w ranges over all nonzero elements of G. An almost difference set is a generalization of a difference set where t=0 or t=v-1. A difference set is said to be a skew Hadamard difference set if G is the disjoint union of D, -D, and {0}.

Almost difference sets have applications in cryptography, coding theory, and code division multiple access (CDMA) communications. In this work, the authors give two constructions of almost difference sets, as well as several results on the equivalence relation of almost difference sets. The first construction generates infinitely many almost difference sets in (GF(q),+) in terms of cyclotomic classes of order 8 in GF(q) for certain values of q. The second construction is in terms of two skew Hadamard difference sets of orders q and q+4, where q and q+4 are prime powers with q=3 (mod 4). The claims made appear to be correct.