Existence of skew Hadamard difference sets and cyclotomic strongly regular graphs on finite fields

2015 Fiscal Year Final Research Report

• Project/Area Number: 25800093
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kumamoto University
• Principal Investigator: MOMIHARA KOJI 熊本大学, 教育学部, 准教授 (70613305)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 組合せデザイン / アソシエーションスキーム / 強正則グラフ / 差集合 / 平方剰余差集合 / 歪アダマール差集合 / 円分強正則グラフ / 有限体

Outline of Final Research Achievements

The objective of this research project is to make clear the existence of skew Hadamard difference sets and cyclotomic strongly regular graphs and their relationship. In previous research, several examples of skew Hadamard difference sets were found, which are inequivalent to the quadratic residue difference sets. We found a nice connection between these two different combinatorial structures. We found that the orders for which such sporadic skew Hadamard difference sets and cyclotomic strongly regular graphs exist coincide. This is a starting point of our research. Indeed, we showed that infinitely many skew Hadamard difference sets can be obtained from cyclotomic strongly regular graphs. Furthermore, we proved that the family includes infinitely many skew Hadamard difference sets inequivalent to the quadratic residue difference sets.

Free Research Field

数物系科学