Algebraic curves for coding theory and their properties

• Project/Area Number: 17K05344
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Shiga University of Medical Science
• Principal Investigator: Kawakita Motoko 滋賀医科大学, 医学部, 准教授 (80467373)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 代数曲線 / 有理点 / 符号理論 / 最大曲線 / Serre上界 / 代数幾何符号 / 有限体

Outline of Final Research Achievements

We find algebraic curves over finite fields with many rational points. For genus 5, we find maximal curves and algebraic curves attaining the Serre bound which are not maximal. For genus 6, we analyze the property of the maximal curves of the Wiman sextics. For genus 7, we find maximal curves and algebraic curves attaining the Serre bound which are not maximal. For genus 10, we find new maximal curves and compute their Jacobian decompositions. Also, we update the database http://www.manypoints.org/.

Academic Significance and Societal Importance of the Research Achievements

ICT 社会基盤の正確性を保つため、誤り訂正符号の理論は不可欠である。 1970年代にゴッパが代数幾何符号を発見した。その理論を用いると、有限体上において多数の有理点をもつ代数曲線から、効率のよい符号が構成できる。本研究の成果は、有限体上において多数の有理点をもつ代数曲線にあり、将来的に実用への貢献が期待できる。また代数曲線論は古くからある純粋数学の研究分野である。存在が知られていない代数曲線を、本研究において具体的に定義方程式を与えたので、数学としても意義がある。

Research Products

• [Int'l Joint Research] University of Perugia(イタリア)
• [Int'l Joint Research] University of Perugia(イタリア)
• [Int'l Joint Research] University of Perugia(イタリア)
• [Int'l Joint Research] University of Perugia(イタリア)
• [Int'l Joint Research] Technical University of Denmark(デンマーク)
• [Journal Article] An Fp2-maximal Wiman sextic and its automorphisms (2021)
  • Author(s): Massimo Giulietti, Motoko Kawakita, Stefano Lia, Maria Montanucci
  • Journal Title: Advances in Geometry Volume: 21 Issue: 4 Pages: 451-461
  • DOI: 10.1515/advgeom-2020-0012
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] New examples of maximal curves with low genus (2020)
  • Author(s): D. Bartoli, M. Giulietti, M. Kawakita, M. Montanucci
  • Journal Title: Finite Fields and Their Applications Volume: 68 Pages: 101744-101744
  • DOI: 10.1016/j.ffa.2020.101744
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Some sextics of genera five and seven attaining the Serre bound (2018)
  • Author(s): 川北素子
  • Journal Title: Lecture Notes in Computer Science Volume: 11321 Pages: 264-271
  • DOI: 10.1007/978-3-030-05153-2_15
  • ISBN: 9783030051525, 9783030051532
  • Peer Reviewed
• [Journal Article] Wiman's and Edge's sextic attaining Serre's bound (2018)
  • Author(s): 川北素子
  • Journal Title: European Journal of Mathematics Volume: 4 Issue: 1 Pages: 330-334
  • DOI: 10.1007/s40879-017-0147-3
  • Peer Reviewed
• [Presentation] Some sextics of genera five and seven attaining the Serre bound (2018)
  • Author(s): 川北素子
  • Organizer: International Workshop on the Arithmetic of Finite Fields
  • Int'l Joint Research
• [Remarks] SUMS -Math-
  • URL: http://www.shiga-med.ac.jp/~kawakita/
• [Remarks] SUMS-Math-
  • URL: http://www.shiga-med.ac.jp/~kawakita/
• [Remarks] SUMS-Math
  • URL: http://www.shiga-med.ac.jp/~kawakita/