Philippine Journal of Science
150 (6B): 1803-1810, December 2021
ISSN 0031 – 7683
Date Received: 21 Jun 2021

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

Jose Maria P. Balmaceda¹ and Benedict M. Estrella²*

¹Institute of Mathematics, College of Science,
University of the Philippines Diliman Quezon City, National Capital Region 1101 Philippines
²Mathematics Department, College of Science, Bulacan State University
Central Luzon Region 3000 Philippines

*Corresponding author: benedict.estrella@bulsu.edu.ph

ABSTRACT

Let q be a prime of the form q = nN + 1 for integers n >= 1 and N > 1 For q < 10 ^ 5 we show that difference sets in the additive group of the field GF(q) are obtained unions of cyclotomic classes of orders N = 12, 20 and 24 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 (127, 63, 31)-difference sets from unions of 18th-cyclotomic classes of GF(127) by constructing six inequivalent (127, 64, 32)-difference sets with zero added from unions of cyclotomic classes of order N = 18