Algebraic theory of codes and designs based on the Terwilliger algebra

• Project/Area Number: 20740003
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: TANAKA Hajime Tohoku University, 大学院・情報科学研究科, 助教 (50466546)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 代数的組合せ論 / アソシエーションスキーム / 符号 / 組合せデザイン / ターウィリガー代数 / Terwilliger代数 / デザイン / 半正定値計画

Research Abstract

I tried to develop a new theory of codes and designs on association schemes based on the (noncommutative) Terwilliger algebra in place of the (commutative) Bose-Mesner algebra. The outcome of the research includes a fundamental contribution to the theory of width and dual width by Brouwer et al., as well as applications of new tools such as semidefinite programming and Leonard pairs.

Research Products

• [Journal Article] Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes (2011)
  • Author(s): 田中太初, 田中利恵
  • Journal Title: European Journal of Combinatorics, Elsevier 32巻,2号 Pages: 155-161
  • Peer Reviewed
• [Journal Article] Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes (2011)
  • Author(s): Hajime Tanaka, Rie Tanaka
  • Journal Title: European Journal of Combinatorics Volume: 32 Pages: 155-161
  • Peer Reviewed
• [Journal Article] Finite Euclidean graphs and Ramanujan graphs (2009)
  • Author(s): 坂内英一, 島袋修, 田中太初
  • Journal Title: Discrete Mathematics, Elsevier 309巻,20号 Pages: 6126-6134
  • Peer Reviewed
• [Journal Article] A bilinear form relating two Leonard systems (2009)
  • Author(s): 田中太初
  • Journal Title: Linear Algebra and its Applications, Elsevier 431巻,10号 Pages: 1726-1739
  • Peer Reviewed
• [Journal Article] Commutative association schemes (2009)
  • Author(s): William J.Martin, 田中太初
  • Journal Title: European Journal of Combinatorics, Elsevier 30巻,6号 Pages: 1497-1525
  • Peer Reviewed
• [Journal Article] New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra (2009)
  • Author(s): 田中太初
  • Journal Title: European Journal of Combinatorics, Elsevier 30巻,3号 Pages: 736-746
  • Peer Reviewed
• [Journal Article] A note on the span of Hadamard products of vectors (2009)
  • Author(s): 田中太初
  • Journal Title: Linear Algebra and its Applications, Elsevier 430巻,2/3号 Pages: 865-867
  • Peer Reviewed
• [Journal Article] Commutative association schemes (2009)
  • Author(s): William J.Martin, Hajime Tanaka
  • Journal Title: European Journal of Combinatorics 30 Pages: 1497-1525
  • Peer Reviewed
• [Journal Article] A bilinear form relating two Leonard systems (2009)
  • Author(s): Hajime Tanaka
  • Journal Title: Linear Algebra and its Applications 431 Pages: 1726-1739
  • Peer Reviewed
• [Journal Article] Finite Euclidean graphs and Ramanujan graphs (2009)
  • Author(s): Eiichi Bannai, Osamu Shimabukuro, Hajime Tanaka
  • Journal Title: Discrete Mathematics 309 Pages: 6126-6134
  • Peer Reviewed
• [Journal Article] A note on the span of Hadamard products of vectors (2009)
  • Author(s): Hajime Tanaka
  • Journal Title: Linear Algebra and its Applications 430 Pages: 865-867
  • Peer Reviewed
• [Journal Article] New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra (2009)
  • Author(s): Hajime Tanaka
  • Journal Title: European Journal of Combinatorics 30 Pages: 736-746
  • Peer Reviewed
• [Journal Article] Commutative association schemes (2009)
  • Author(s): William J. Martin, Hajime Tanaka
  • Journal Title: European Journal of Combinatorics (印刷中)
  • Peer Reviewed
• [Presentation] Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2011)
  • Author(s): 田中太初
  • Organizer: Mini Workshop
  • Place of Presentation: Texas 大学Brownsville 校,Brownsville, Texas, 米国
  • Year and Date: 2011-02-11
• [Presentation] Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2011)
  • Author(s): Hajime Tanaka
  • Organizer: Mini Workshop
  • Place of Presentation: University of Texas at Brownsville,米国
  • Year and Date: 2011-02-11
• [Presentation] Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2011)
  • Author(s): 田中太初
  • Organizer: Linear Algebra Techniques in Combinatorics/Graph Theory
  • Place of Presentation: The Banff Centre, Banff, Alberta, カナダ
  • Year and Date: 2011-02-02
• [Presentation] Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2011)
  • Author(s): Hajime Tanaka
  • Organizer: Linear Algebra Techniques in Combinatorics/Graph Theory
  • Place of Presentation: The Banff Centre, Banff, Alberta, Canada
  • Year and Date: 2011-02-02
• [Presentation] A note on super Catalan numbers (2011)
  • Author(s): 田中太初
  • Organizer: Combinatorics Seminar
  • Place of Presentation: Wisconsin 大学Madison 校, Madison, WI,米国
  • Year and Date: 2011-01-24
• [Presentation] A note on super Catalan numbers (2011)
  • Author(s): Hajime Tanaka
  • Organizer: Combinatorics Seminar
  • Place of Presentation: University of Wisconsin-Madison,米国
  • Year and Date: 2011-01-24
• [Presentation] Subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2010)
  • Author(s): 田中太初
  • Organizer: Combinatorics Seminar
  • Place of Presentation: Wisconsin 大学Madison 校, Madison, WI, 米国
  • Year and Date: 2010-10-25
• [Presentation] Subsets with minimal width and dual width in Q-polynomial distance-regular graphs (2010)
  • Author(s): Hajime Tanaka
  • Organizer: Combinatorics Seminar
  • Place of Presentation: University of Wisconsin-Madison,米国
  • Year and Date: 2010-10-25
• [Presentation] The Terwilliger algebra of a graph (2009)
  • Author(s): 田中太初
  • Organizer: 松江セミナー
  • Place of Presentation: 松江大学
  • Year and Date: 2009-09-09
• [Presentation] 具体例に於けるTerwilliger代数の既約加群の計算 (2009)
  • Author(s): 田中太初
  • Organizer: 小研究集会「有限幾何とその周辺」
  • Place of Presentation: 東京女子大学
  • Year and Date: 2009-08-07
• [Presentation] 具体例に於ける Terwilliger 代数の既約加群の計算 (2009)
  • Author(s): 田中太初
  • Organizer: 小研究集会「有限幾何とその周辺」
  • Place of Presentation: 東京女子大学
  • Year and Date: 2009-08-07
• [Presentation] Z_4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets (2009)
  • Author(s): 田中太初
  • Organizer: 夏の学校「Designs and Codes」
  • Place of Presentation: ヒルズサンピア山形
  • Year and Date: 2009-06-23
• [Presentation] Z_4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets (2009)
  • Author(s): 田中太初
  • Organizer: 夏の学校「Designs and Codea」
  • Place of Presentation: ヒルズサンピア山形
  • Year and Date: 2009-06-23
• [Presentation] Dual tight cometric association schemes (2009)
  • Author(s): 須田庄, 田中太初, 田中利恵
  • Organizer: AMS 2009 Eastern Section Meeting
  • Place of Presentation: Worcester 工芸大学, Worcester, MA, 米国
  • Year and Date: 2009-04-26
• [Presentation] Dual tight cometric association schemes (2009)
  • Author(s): 田中太初
  • Organizer: AMS 2009 Eastern Section Meeting
  • Place of Presentation: Worcester Polytechnic Institute, USA
  • Year and Date: 2009-04-26
• [Presentation] A bilinear form relating two Leonard systems (2009)
  • Author(s): 田中太初
  • Organizer: Combinatorics Seminar
  • Place of Presentation: Wisconsin 大学Madison 校,Madison, WI, 米国
  • Year and Date: 2009-04-20
• [Presentation] A bilinear form relating two Leonard systems (2009)
  • Author(s): 田中太初
  • Organizer: Combinatorics Seminar
  • Place of Presentation: Department of Mathematics, University of Wisconsin, USA
  • Year and Date: 2009-04-20
• [Presentation] Dual-tight cometric association schemes (2009)
  • Author(s): 須田庄, 田中太初
  • Organizer: ミニ集会「代数的組合せ論」
  • Place of Presentation: 九州大学
  • Year and Date: 2009-03-18
• [Presentation] Dual-tight com etric association schemes (2009)
  • Author(s): 田中太初
  • Organizer: ミニ集会「代数的組合せ論」
  • Place of Presentation: 福岡
  • Year and Date: 2009-03-18
• [Presentation] The nucleus of a Q-polynomial distance-regular graph (2008)
  • Author(s): 田中太初
  • Organizer: The 5th Japan-Korea Workshop on Algebra and Combinatorics
  • Place of Presentation: 九州大学
  • Year and Date: 2008-09-19
• [Presentation] The nucleus of a Q-polynomial distance-regular graph (2008)
  • Author(s): Hajime Tanaka
  • Organizer: The 5th Japan-Korea Workshop on Algebra and Combinatorics
  • Place of Presentation: 福岡
  • Year and Date: 2008-09-19
• [Presentation] A bilinear form relating two Leonard pairs and its applications (2008)
  • Author(s): 田中太初
  • Organizer: Geometric and Algebraic Combinatoris 4, Hotel Conferentiecentrum Boschoord
  • Place of Presentation: Oisterwijk, オランダ
  • Year and Date: 2008-08-20
• [Presentation] A bilinear form relating two Leonard pairs and its applications (2008)
  • Author(s): Hajime Tanaka
  • Organizer: Geometric and Algebraic Combinatoris 4
  • Place of Presentation: Oisterwijk, The Netherlands
  • Year and Date: 2008-08-20