Difference Sets from Unions of Cyclotomic Classes of Orders 12, 20, and 24

Abstract

Let 𝒒𝒒 be a prime of the form 𝒒𝒒 = 𝒏𝒏𝒏𝒏 + 𝟏𝟏 for integers 𝒏𝒏 ≥ 𝟏𝟏 and 𝑵𝑵 > 𝟏𝟏. For 𝒒𝒒 < 𝟏𝟏𝟏𝟏𝟓𝟓, we show that difference sets in the additive group of the field 𝐆𝐆𝑭𝑭(𝒒𝒒) are obtained from unions of cyclotomic classes of orders 𝑵𝑵 = 𝟏𝟏𝟏𝟏, 𝟐𝟐𝟐𝟐, and 𝟐𝟐𝟐𝟐 and determine all such unions using a computer search. We then determine if the difference sets are equivalent to known cyclotomic or modified cyclotomic quadratic, quartic, sextic, or octic difference sets or their complements. This fills the gaps in the literature on the existence of difference sets from unions of cyclotomic classes for the specified orders. In addition, we extend Baumert and Fredricksen’s 1967 work on the construction of all inequivalent (𝟏𝟏𝟏𝟏𝟏𝟏, 𝟔𝟔𝟔𝟔, 𝟑𝟑𝟑𝟑)-difference sets from unions of 𝟏𝟏𝟏𝟏𝒕𝒕𝒕𝒕- cyclotomic classes of 𝑮𝑮𝑮𝑮(𝟏𝟏𝟏𝟏𝟏𝟏) by constructing six inequivalent (𝟏𝟏𝟏𝟏𝟏𝟏, 𝟔𝟔𝟔𝟔, 𝟑𝟑𝟑𝟑)-difference sets with zero added from unions of cyclotomic classes of order 𝑵𝑵 = 𝟏𝟏𝟏𝟏.