Note on divisible difference sets from Galois rings GR(9,n)

TL;DR

This paper introduces a new method for constructing divisible difference sets in certain algebraic structures using Galois rings, contingent on the existence of skew Hadamard difference sets in finite fields.

Contribution

It provides a novel construction of divisible difference sets in ${f Z}_9^n$ leveraging Galois rings and the existence of skew Hadamard difference sets.

Findings

New construction of divisible difference sets in ${f Z}_9^n$

Connection established between Galois rings and difference sets

Conditional on skew Hadamard difference sets in ${f F}_{3^n}$

Abstract

In this note, we give a new construction of divisible difference sets in ${\Bbb Z}_{9}^n$ using Galois ring $GR(3^2,n)$ under the assumption of the existence of skew Hadamard difference sets in ${\Bbb F}_{3^n}$.