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

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03544
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Japan 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
• Keywords: 代数幾何符号 / ベクトル束 / adelic曲線

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.

Free Research Field

代数幾何学

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。