A study of combinatorics over Galois rings

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540013
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Woman's Christian University (2014); Kanazawa University (2012-2013)
• Principal Investigator: YAMADA Mieko 東京女子大学, 現代教養学部, 研究員 (70130226)
• Co-Investigator(Kenkyū-buntansha): MOMIHARA Koji 熊本大学, 教育学部, 講師 (70613305)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: ガロア環 / 差集合 / ガウス和 / Hadamard行列 / difference family / 符号

Outline of Final Research Achievements

We gave a new approach and a new perspective on the study of combinatorics, that is,recognizing a combinatorial structure over finite fields and Galois rings as an image of a subset of a local　field by natural projections. We obtained an infinite family of Menon-Hadamard difference sets over Galois rings of characteristic a power of 2 from a subset of 2-adic field by natural projections, which has a　nested structure. Furthermore we showed that there exist difference families and Hadamard matrices over Galois rings of　characteristic 4. For odd characteristics, we constructed a decoding algorithm of BCH codes with at most 2 errors. For a characteristic a square of a prime and an　extension degree 2, we constructed difference families and Hadamard matrices.
In addition to the above results, we constructed Cayley graphs and their lifting and provided a new insight of the classification of skew Hadamard matrices over finite fields.

Free Research Field

組合せ数学