CONSTRUCTION OF HIGHER DIMENSIONAL ALGEBRAIC-GEOMETRIC CODES BASED ON VECTOR BUNDLES

• Project/Area Number: 18540043
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Japan Women's University
• Principal Investigator: NAKASHIMA Toru Japan Women's University, 理学部, 教授 (20244410)
• Co-Investigator(Kenkyū-buntansha): TERAO Hiroaki 北海道大学, 大学院・理学研究院, 教授 (90119058) ; MARTIN Guest 首都大学東京, 都市教養学部, 教授 (10295470)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,240,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥540,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 代数幾何符号 / ベクトル束 / 相対的Reed-Muller符号 / 代数幾何 / 誤り訂正符号

Research Abstract

情報を通信する過程では通常雑音によって誤りが生じる。これらの誤りを訂正するための数学的手法は誤り訂正符号とよばれ、コンパクトディスクの再生から火星探査機との交信に至るまで広く用いられている。中でも現在最も性能の良い符号は、多項式の共通零点が表す図形である代数多様体の幾何学を用いた代数幾何符号である。当研究では、ベクトル束とよばれる対象を用いる事によって高次元の代数多様体から新しいタイプの誤り訂正符号を構成し、その性質を解明する事に成功した。

Research Products

• [Journal Article] Minimum distance of relative Reed-Muller codes (2009)
  • Author(s): T.Nakashima
  • Journal Title: Appl.Algebra Eng.Commun.Comput 20 Pages: 123-132
  • Peer Reviewed
• [Journal Article] Totally free arrangements of hyperplanes (2009)
  • Author(s): T.Abe, H.Terao,M. Yoshinaga
  • Journal Title: Proc.Amer.Math.Soc 137 Pages: 1405-1410
  • NAID: 120006459611
  • Peer Reviewed
• [Journal Article] Brill-Noether problems in higher dimensions (2008)
  • Author(s): T.Nakashima
  • Journal Title: Forum Math 20 Pages: 145-161
  • Peer Reviewed
• [Journal Article] Existence of stable vector bundles on Calabi-Yau manifolds (2008)
  • Author(s): T.Nakashima
  • Journal Title: RIMS Kokyuroku Bessatsu B9 Pages: 1977-1983
  • Peer Reviewed
• [Journal Article] Periodicity of hyperplane arrangements with integral coefficients modulo positive integers (2008)
  • Author(s): H.Kamiya, A.Takemura, H.Terao
  • Journal Title: J.Algebraic Combin 27 Pages: 317-330
  • NAID: 120006459543
  • Peer Reviewed
• [Journal Article] The Euler multiplicity and additiondeletion theorems for multiarrangements (2008)
  • Author(s): T.Abe, H.Terao, M.Wakefield
  • Journal Title: J.London Math.Soc (2)77 Pages: 335-348
  • Peer Reviewed
• [Journal Article] Brill-Noether problems in higher dimensions (2008)
  • Author(s): T. Nakashima
  • Journal Title: Forum Math. 20 Pages: 145-161
  • Peer Reviewed
• [Journal Article] Existence of stable bundles on Calabi-Yau manifolds (2008)
  • Author(s): T. Nakashima
  • Journal Title: RIMS Kokyuroku Bessatsu B9 Pages: 1977-1983
  • Peer Reviewed
• [Journal Article] Strong Bogomolov inequality for stable vector bundles (2007)
  • Author(s): T.Nakashima
  • Journal Title: J.Geom.Phys 57 Pages: 1977-1983
  • Peer Reviewed
• [Journal Article] The characteristic polynomial of a multiarrangements (2007)
  • Author(s): T.Abe, H.Terao, M.Wakefield
  • Journal Title: Adv.Math 215 Pages: 825-838
  • NAID: 120006459522
  • Peer Reviewed
• [Journal Article] Chambers of arrangements of hyperplanes and Arrow's impossibility theorem (2007)
  • Author(s): H.Terao
  • Journal Title: Adv.Math 214 Pages: 366-378
  • NAID: 120000955425
  • Peer Reviewed
• [Journal Article] Strong Bogomolov inequality for stable vector bundles (2007)
  • Author(s): T.Nakashima
  • Journal Title: Journal of Geometry and Physics 57 Pages: 1977-1983
  • Peer Reviewed
• [Journal Article] Chambers of arrangements of hyperplanes and Arrow's impossibiity theorem (2007)
  • Author(s): H.Terao
  • Journal Title: Advances in Mathematics 214 Pages: 366-378
  • Peer Reviewed
• [Journal Article] The characteristic polynomial of a multiarrangement (2007)
  • Author(s): T.Abe, H.Terao, M.Wakefield
  • Journal Title: Advances in Mathematics 215 Pages: 825-838
  • NAID: 120006459522
  • Peer Reviewed
• [Journal Article] Error-correcting codes on projective bundles (2006)
  • Author(s): T.Nakashima
  • Journal Title: Finite Fields and Their Appl 12 Pages: 222-231
  • Peer Reviewed
• [Journal Article] Error-correcting codes on projective bundles (2006)
  • Author(s): T.Nakashima
  • Journal Title: Finite Fields and Their Applications 12 Pages: 222-231
• [Journal Article] Arrangements and ranking patterns (2006)
  • Author(s): H.Kamiya, P.Orlik, A.Takemura, H.Terao
  • Journal Title: Annals of combinatorics 10 Pages: 219-235
• [Journal Article] Introduction to homological geometry I (2006)
  • Author(s): M.A.Gueset
  • Journal Title: Integrable systems, geometry, and topology, AMS/IP Stud.Adv.Math. 36 Pages: 123-150
• [Presentation] Existence of stable bundles on Calabi-Yau manifolds (2007)
  • Author(s): 中島徹
  • Organizer: 高次元代数多様体とベクトル束
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-07-04
• [Presentation] Existence of stable bundles on Calabi-Yau manifolds (2007)
  • Author(s): 中島 徹
  • Organizer: 研究集会「高次元代数多様体とベクトル束」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-07-04
• [Remarks] 中島徹Who's Who in the World, 26 th Editionに登録