Menon-hadamard difference sets obtained from a local field by natural projections

Abstract

We know there exists a family of Menon-Hadamard difference sets over Galois rings of characteristic of an even power of 2 and of an odd extension degree, which has a nested structure. The projective limit of these Menon-Hadamard difference sets is a non-empty subset of a valuation ring of a local field. Conversely, does there exist a subset of a local field whose image by the natural projection always gives a difference set over a Galois ring? We will show an answer to this problem. A family of Menon-Hadamard difference sets is obtained from a subgroup of a valuation ring of a local field by the natural projections and it also has a nested structure. The formal group and the p-adic logarithm function serve an important role to the construction.