Almost Difference Sets, Normally Regular Digraphs and Cyclotomic Schemes from Cyclotomy of Order Twelve

TL;DR

This paper constructs infinite families of almost difference sets and normally regular graphs using cyclotomic classes of order twelve, analyzing their properties through cyclotomy in finite fields.

Contribution

It introduces new almost difference sets from cyclotomic classes of order twelve and computes their intersection numbers and character tables.

Findings

Constructed infinite families of almost difference sets.

Analyzed intersection numbers and character tables of associated schemes.

Showed that certain unions of cosets form almost difference sets.

Abstract

Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the multiplicative cyclic subgroup $C$ of index twelve nor $C\cup \{0\}$ forms an almost difference set, but a union of cosets of $C$ provides us an almost difference set. We also calculate the intersection numbers and character tables of cyclotomic association schemes of class two, three and four obtained from these fields.