Designs, Codes and Cryptography, 28, 75–91, 2003  C 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three K. T. ARASU ∗ Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, U.S.A. KEVIN J. PLAYER † Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, U.S.A. Communicated by: D. Jungnickel Received December 8, 2000; Revised November 16, 2001; Accepted November 29, 2001 Abstract. We construct a new family of cyclic difference sets with parameters ((3 d - 1)/2,(3 d-1 - 1)/2, (3 d-2 - 1)/2) for each odd d . The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti’s hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets. Keywords: difference set, characters, group rings, Gauss sums, Jacobi sums AMS Classification: 05B10, 11L05 1. Introduction Let G be an abelian group order v and D be a subset of G with cardinality k . D is said to be a (v, k , λ) difference set if the multiset (g 1 g -1 2 : g 1 , g 2 ∈ D, g 1  = g 2 ) contains every nonidentity element of G exactly λ times. An easy counting argument shows that k (k - 1) = λ(v - 1). Define n = k - λ; n is called the order of D. We say that D is cyclic (resp. abelian) if G is cyclic (resp. abelian). Let ZG denote the group ring of G over Z. We identify each subset S of G with the group ring element S = ∑ x ∈S X . For A = ∑ g∈G a g g ∈ ZG and any integer t , we define A (t ) = ∑ g∈G a g g t . With these notations, the difference set condition of D ⊆ G can be rewritten as DD (-1) = n + λG in ZG (n = n ∗ 1 ∈ ZG) Difference sets are often studied using character theory. Let G ∗ be the group of characters χ (homomorphisms from the group G to the field of complex numbers C). The principal ∗ The work was partially supported by NSF grant CCR 9814106 and by NSA grant 904-01-1-0041. † The work was partially supported by an REU grant from NSF.