New constructions of designs, graphs and codes over finite fields based on finite geometry and algebraic methods

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K14236
• 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, 大学院先端科学研究部(理), 准教授 (70613305)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: 差集合 / 強正則グラフ / Griesmer符号 / アダマール行列 / アソシエーションスキーム / 有限幾何 / ガウス周期

Outline of Final Research Achievements

In past researches, some constructions for combinatorial objects such as designs, graphs and codes have been presented based on finite geometry of small dimension and cyclotomy of small index over finite fields. However, because of the difficulty of computations of the size of hyperplane sections and Gauss sums, the classes of such objects found before has been limited. In this research, we found new approaches to treat geometric objects of large dimension and compute Gauss sums of large index using a combination of actions of finite groups and field extensions. In particular, we succeeded to give new families of skew Hadamard difference sets inequivalent to known ones, skew Hadamard difference families with new parameters, Hadamard matrices of new order, and new strongly regular graphs based on three-valued Gauss periods. Thus, we generalized known construction theories for designs, graphs and codes over finite fields.

Free Research Field

離散数学, 組合せ論

Academic Significance and Societal Importance of the Research Achievements

デザイン・グラフ・符号はそれぞれ, 統計学・ネットワーク・情報通信に応用され, 我々の日常生活の背後にある重要な離散構造である. 一方, その存在性に関しては, 未解決な部分が多く組合せ論における重要な研究課題である. 特に, 本研究は, 有限体上の差集合・強正則グラフ・Griesmer符号と呼ばれるデザイン・グラフ・符号について, 有限幾何・整数論・群論的手法を組み合わせた新手法を考案し, 多くのパラメータで存在性が未知であった上記組合せ構造の存在性の解明を行った.