Construction of Difference Sets from Unions of Cyclotomic Classes of Order N=14

Abstract

Let G be an additive group of order v, D be a non-empty proper k-subset of G, and λ be any integer. Then D is a (v, k, λ) - difference set if every nonzero element of the group can be expressed as a difference d₁ - d₂ of elements of D in exactly λ ways. Let q be a prime of the form q = nN + 1 for integers n>1 and N>1. For q<1000, this study shows the construction of difference sets in the additive group of the field GF(q) from unions of cyclotomic classes of order N = 14 using a computer search. The construction consisted of computer programs derived from the definitions and theorems on difference sets using Python. The results revealed that only the union of seven cyclotomic classes such as C₀^{(14, q)} ∪ C₂^{(14, q)} ∪ C₄^{(14, q)} ∪ C₆^{(14, q)} ∪ C₈^{(14, q)} ∪ C₁₀^{(14, q)} ∪ C₁₂^{(14, q)} forms a quadratic cyclotomic difference set. Similarly, this union together with zero forms a difference set equivalent to the modified quadratic cyclotomic difference sets.