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Research on the algebraic-geometric codes based on adelic vector bundles

Research Project

Project/Area Number 20K03544
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionJapan Women's University

Principal Investigator

NAKASHIMA Tohru  日本女子大学, 理学部, 教授 (20244410)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords代数幾何符号 / ベクトル束 / adelic曲線 / adelic符号 / Seshadri定数 / adelic ベクトル束 / Arakelov幾何学
Outline of Research at the Start

データを送信する過程で生じた誤りを訂正し、元の情報を復元するための誤り訂正符号は現代のIT社会に欠かせない技術である。本研究では、数と図形を共通の枠組で取り扱うことを可能にするadelic幾何学と呼ばれる理論を用いて、従来知られている既知の様々な符号を統一する新しいタイプの符号(adelic符号)を導入する。又、その性質を調べることによって従来より高い性能をもつadelic符号を構成することを目指す。

Outline of Final Research Achievements

The error-correcting code is an indispensable technology in the modern transmission of information. Among them, the algebraic geometric code has very high ability of error correction. In this project, we made a research on a novel type of algebraic geometric code called adelic code which is based on the adelic curves and vector bundles on them. As a result, under suitable assumptions we could determine the parameters of these adelic codes such as minimum distances or dimensions and investigated detailed properties of them in some concrete examples. We also clarified the relation between asymptotic minimal slopes of vector bundles in positive characteristic and adelic codes.

Academic Significance and Societal Importance of the Research Achievements

当研究によって、有限体上の代数多様体と算術的多様体から定義される異なるタイプの代数幾何符号達をadelic符号の観点から統一的に理解するための枠組みを与えることができた。この結果は代数幾何符号の対象を大幅に拡大していくことを可能にするという点で重要な意義をもつものと考えられる。また、当研究ではasymptotic minimal slopeという正標数のベクトル束に固有の不変量を用いて符号のパラメーターを評価する新しい手法を開発できた。今後この手法を更に発展させることによって従来より高い性能をもつ誤り訂正符号が構成できれば情報通信の分野への応用が期待される。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (2 results)

All 2024 2022

All Journal Article (2 results) (of which Peer Reviewed: 2 results)

  • [Journal Article] AG codes on flag bundles over a curve2024

    • Author(s)
      Tohru Nakashima
    • Journal Title

      Finite Fields and Their Applications

      Volume: 95 Pages: 102392-102392

    • DOI

      10.1016/j.ffa.2024.102392

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Cohomology bound and Chern class inequalities for stable sheaves on a smooth projective variety2022

    • Author(s)
      Tohru Nakashima
    • Journal Title

      Indian Journal of Pure and Applied Mathematics

      Volume: 12 Issue: 3 Pages: 789-796

    • DOI

      10.1007/s13226-022-00297-8

    • Related Report
      2022 Research-status Report
    • Peer Reviewed

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Published: 2020-04-28   Modified: 2025-01-30  

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