Almost Difference Sets from Unions of Cyclotomic Classes of Order 14

TL;DR

This paper explores the construction of almost difference sets from unions of cyclotomic classes of order 14 in finite fields, using computational search to identify new sets with potential applications in coding theory and sequence design.

Contribution

It introduces a new construction method for almost difference sets based on cyclotomy of order 14, expanding the known classes of such sets.

Findings

Identified new almost difference sets from unions of cyclotomic classes of order 14.

Developed an exhaustive computational approach to test for almost difference sets.

Determined equivalence classes among the generated sets.

Abstract

Almost difference sets have emerged as a fascinating and important area of research as they can produce functions with optimal nonlinearity, cyclic codes, and binary sequences with optimal autocorrelation. This study aims to investigate the existence of almost difference sets from the union of suitable cyclotomic classes of order 14 (with and without the residue zero) of the finite field $GF(q)$, where $q$ is a prime of the form $q=14n+1$ for positive integers $n\geq 1$ and $q<1000$. The construction utilized an exhaustive computer search using Python. The method computes the unions of two classes up to thirteen classes and tests the existence of almost difference sets. The equivalence of the generated almost difference sets with the same parameters is also determined. The findings will contribute to the literature with a new construction of almost difference sets via cyclotomy of order 14.