Constructions of Low Rank Relative Difference Sets in 2-Groups Using Galois Rings

In this paper, we first determine the exact values of s(2, t), a function defined by X.-D. Hou [preprint], which depends on the structure of GR(4, t). This result gives the exact range of parameters of the partial difference sets in t4 constructed by Chen et al. (J. Combin. Theory Ser. A76 (1996), 179–196) and Hou (preprint). Second, we construct a new family of (22t, 2t, 22t, 2t) relative difference sets in abelian 2-groups of rank as low as t+ . This is a significant reduction over the previous minimum of 2t found in Davis (Discrete Math.103, (1992), 7–15).