New constructions of Hadamard matrices based on difference familiesand their generalizations

• Project/Area Number: 23840032
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kumamoto University
• Principal Investigator: MOMIHARA Koji 熊本大学, 教育学部, 講師 (70613305)
• Research Collaborator: YAMADA Mieko 金沢大学, 数物科学系, 教授 (70130226)
• Project Period (FY): 2011 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 組合せ論 / 応用数学 / 組合せデザイン / アダマール行列 / 差集合族 / 差集合 / 強正則グラフ / アダマール型差集合 / 有限体 / ガロア環 / 相対ガウス和 / ガウス和 / 平方剰余型差集合

Research Abstract

The objective of this research project is to propose new constructions of Hadamardmatrices. Hadamard matrices have several applications in the research areas ofCommunications, Statistics, etc. In the past researches, several constructions ofHadamard matrices have been presented based on difference families, which were obtainedfrom difference sets in finite fields. In our research, by generalizing the knownconstructions of difference families using difference sets in finite fields to those incommutative rings, we extended both of the theories of Hadamard matrices and differencefamilies. In particular, by introducing a new concept generalizing ordinary differencefamilies, we found a new infinite series of difference families in Galois rings and anew construction of Hadamard matrices using the new ones. Furthermore, we investigatedthe linkage between Hadamard matrices, Hadamard difference sets, and strongly regulargraphs.

Research Products

• [Journal Article] Strongly regular Cayleygraphs, skew Hadamard difference sets, andrationality of relative Gauss sums (2013)
  • Author(s): Koji Momihara
  • Journal Title: European Journal of Combinatorics Volume: 34巻 Pages: 18-18
  • Peer Reviewed
• [Journal Article] Three-classassociation schemes from cyclotomy、Journal of Combinatorial Theory (2013)
  • Author(s): Tao Feng、Koji Momihara
  • Journal Title: Series A Volume: 120巻 Pages: 14-14
  • Peer Reviewed
• [Journal Article] Strongly regular Cayley graphs, skew Hadamard difference sets, and rationality of relative Gauss sums (2013)
  • Author(s): K. Momihara
  • Journal Title: European Journal of Combinatorics Volume: 34 Issue: 4 Pages: 706-723
  • DOI: 10.1016/j.ejc.2012.10.006
  • Peer Reviewed
• [Journal Article] Three-class association schemes from cyclotomy (2013)
  • Author(s): K. Momihara
  • Journal Title: Journal of Combinatorial Theory, Series A Volume: 120 Issue: 6 Pages: 1202-1215
  • DOI: 10.1016/j.jcta.2013.03.002
  • Peer Reviewed
• [Journal Article] A decomposition of the 2-designformed by the planes in AG(2n,3) (2012)
  • Author(s): Koji Momihara、Miwako Mishima、MasakazuJimbo
  • Journal Title: Finite Fields and Their Applications Volume: 18巻 Pages: 956-970
  • Peer Reviewed
• [Journal Article] A decomposition of the 2-design formed by the planes in AG(2n,3) (2012)
  • Author(s): Koji Momihara, Miwako Mishima, Masakazu Jimbo
  • Journal Title: Finite Fields and Their Applications Volume: Vol.18 Issue: 5 Pages: 956-970
  • DOI: 10.1016/j.ffa.2012.04.001
  • Peer Reviewed
• [Journal Article] Amorphous association scheme に 関するIvanov 予想の反例の一般化 (2011)
  • Author(s): 籾原幸二 、 Tao Feng 、 Qing Xiang
  • Journal Title: 応用数学合同研究集会報告集 Pages: 66-67
• [Journal Article] Amorphous association schemeに関するIvanov予想の反例の一般化 (2011)
  • Author(s): 籾原幸二, Tao Feng, Qing Xiang
  • Journal Title: 応用数学合同研究集会報告集 Pages: 66-67
• [Presentation] Lifting construction ofstrongly regular Cayley graphs in F_q (2012)
  • Author(s): 籾原幸二
  • Organizer: The 2nd Japan-Taiwan Conference onCombinatorics and its Applications
  • Place of Presentation: 名古屋大学(愛知)
  • Year and Date: 2012-11-12
• [Presentation] Strongly regular Cayleygraphs and rationality of relative Gausssums (2012)
  • Author(s): Koji Momihara
  • Organizer: Combinatorics 2012
  • Place of Presentation: Hotel Gio(イタリア・ペルージャ)
  • Year and Date: 2012-09-14
• [Presentation] 強正則グラフ Cay(F_q,D)と相対ガウス和の有理性について (2012)
  • Author(s): 籾原幸二
  • Organizer: RIMS 研究集会「デザイン、 符号、 グラフおよびその周辺」
  • Place of Presentation: 京都大学(京都)
  • Year and Date: 2012-07-19
• [Presentation] Recent progress on cyclotomicstrongly regular graphs and skew Hadamard difference sets (2012)
  • Author(s): 籾原幸二
  • Organizer: 研究集会「有限体とそれに関連する代数的組合せ論」
  • Place of Presentation: 神戸学院大学(兵庫)
  • Year and Date: 2012-03-03
• [Presentation] Recent progress on cyclotomic strongly regular graphs and skew Hadamard difference sets (2012)
  • Author(s): 籾原幸二
  • Organizer: 研究集会「有限体とそれに関連する代数的組合せ論」
  • Place of Presentation: 神戸学院大学(兵庫県)
  • Year and Date: 2012-03-03
• [Presentation] Lifting constructions ofstrongly regular graphs and associationschemes in F_q (2012)
  • Author(s): 籾原幸二
  • Organizer: RIMS 研究集会「有限群とその表現,頂点作用素代数,代数的組合せ論の研究」
  • Place of Presentation: 京都大学(京都)
  • Year and Date: 2012-01-08
• [Presentation] Lifting constructions ofstrongly regular Cayley graphs (2012)
  • Author(s): 籾原幸二
  • Organizer: 研究集会「代数的グラフ理論、スペクトラルグラフ理論および周辺領域」
  • Place of Presentation: 名古屋大学(愛知)
  • Year and Date: 2012-01-06
• [Presentation] (2011)
  • Author(s): 籾原幸二、Tao Feng、Qing Xiang
  • Organizer: 2011年度応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀)
  • Year and Date: 2011-12-16
• [Presentation] Amorphous association schemeに関するIvanov予想の反例の一般化 (2011)
  • Author(s): 籾原幸二, Tao Feng, Qing Xiang
  • Organizer: 2011年度応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀県)
  • Year and Date: 2011-12-16
• [Presentation] Supplementary divisibledifference sets over Galois rings ofcharacteristic 4 from generalizedSzekeres's construction (2011)
  • Author(s): 籾原幸二
  • Organizer: 日本数学会秋季総合分科会応用数学分科会
  • Place of Presentation: 信州大学(長野)
  • Year and Date: 2011-09-29
• [Presentation] Supplementary divisible difference sets over Galois rings of characteristic 4 from generalized Szekeres's construction (2011)
  • Author(s): 籾原幸二
  • Organizer: 日本数学会秋季総合分科会応用数学分科会
  • Place of Presentation: 信州大学(長野県)
  • Year and Date: 2011-09-29
• [Presentation] 強正則グラフCay(F_q,D)と相対ガウス和の有理性について
  • Author(s): 籾原幸二
  • Organizer: RIMS 研究集会「デザイン、符号、グラフおよびその周辺」
  • Place of Presentation: 京都(京都大学)
  • Invited
• [Presentation] Strongly regular Cayley graphs and rationality of relative Gauss sums
  • Author(s): 籾原幸二
  • Organizer: Combinatorics 2012
  • Place of Presentation: イタリア(ペルージャ)
• [Presentation] Lifting construction of strongly regular Cayley graphs in F_q
  • Author(s): 籾原幸二
  • Organizer: The 2nd Japan-Taiwan Conference on Combinatorics and its Applications
  • Place of Presentation: 愛知(名古屋大学)
• [Presentation] Lifting constructions of strongly regular Cayley graphs
  • Author(s): 籾原幸二
  • Organizer: 研究集会「代数的グラフ理論、スペクトラルグラフ理論および周辺領域」
  • Place of Presentation: 愛知(名古屋大学)
• [Presentation] Lifting constructions of strongly regular graphs and association schemes in F_q
  • Author(s): 籾原幸二
  • Organizer: RIMS 研究集会「有限群とその表現，頂点作用素代数，代数的組合せ論の研究」
  • Place of Presentation: 京都(京都大学)
  • Invited
• [Remarks]
  • URL: http://combin-math-kumamoto.jp/
• [Remarks]
  • URL: http://www.educ.kumamoto-u.ac.jp/~momihara/
• [Remarks] 熊本大学 教育学部数学科 応用数学研究室
  • URL: http://combin-math-kumamoto.jp/
• [Remarks] Web page of Koji Momihara
  • URL: http://www.educ.kumamoto-u.ac.jp/~momihara/
• [Remarks]
  • URL: http://combin-math-kumamoto.jp/
• [Remarks] または
  • URL: http://www.educ.kumamoto-u.ac.jp/~momihara/