Difference sets over Galois rings with odd extension degrees and characteristic an even power of 2

Abstract

We construct an infinite family of (2^ns, 2^{ns/2 -1}(2^ns/2−1), 2^{ns/2 -1}(2^{ns/2 -1} −1)) difference sets over a Galois ring GR(2ⁿ, s) with characteristic an even power n of 2 and an odd extension degree s. It makes a chain of difference sets preserving the structures when n increases and s is fixed. We introduce a new operation into GR(2ⁿ, s). The Gauss sum associated with the multiplicative character defined by the subgroup with respect to the new operation plays an important role in the construction.