On Some New Almost Difference Sets Constructed from Cyclotomic Classes of Order 12

Abstract

Almost Difference Sets have extensive applications in coding theory and cryptography. In this study, we introduce new constructions of Almost Difference Sets derived from cyclotomic classes of order 12 in the finite field GF(q), where q is a prime satisfying the form q=12n+1 for positive integers n ≥ 1 and q < 1000. We show that a single cyclotomic class of order 12 (with and without zero) can form an almost difference set. Additionally, we successfully construct almost difference sets using unions of cyclotomic classes of order 12, both for even and odd values of n. To accomplish this, an exhaustive computer search employing Python was conducted. The method involved computing unions of two cyclotomic classes up to eleven classes and assessing the presence of almost difference sets. Finally, we classify the resulting almost difference sets with the same parameters up to equivalence and complementation.