Divisible difference families from Galois rings GR(4,n) and Hadamard matrices

TL;DR

This paper introduces new constructions of difference families from Galois rings and finite fields, and presents a novel method for constructing symmetric Hadamard matrices using divisible difference families.

Contribution

It generalizes Szekeres's difference families, provides new examples in finite fields, and introduces a new construction method for symmetric Hadamard matrices.

Findings

Existence of infinite divisible difference families in Galois rings

New difference families in multiplicative subgroups of finite fields

A novel construction method for symmetric Hadamard matrices

Abstract

We give a new construction of difference families generalizing Szekeres's difference families \cite{Sze}. As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring GR(4,n). Furthermore, we obtain a new construction method of symmetric Hadamard matrices by using divisible difference families and a new array.