Research on the algebraic-geometric codes defined from restriction of vector bundles to divisors

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05111
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• 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): 2016-04-01 – 2020-03-31
• Keywords: 代数幾何符号 / ベクトル束 / 因子

Outline of Final Research Achievements

In the modern society the error-correcting codes, which correct the errors which occurred in the process of transmitting information, is an indispensable technology. The purpose of the present research was to elucidate the properties of the general Savin code, which is an algebraic-geometric code defined by the restriction of vector bundles to divisors. As a result of the research, we could determine the parameters of the general Savin codes on certain algebraic surfaces. Furthermore we solved the existence problem of semistable sheaves on projective threefolds and proved a Bogomolov-Gieseker type inequality.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

従来知られていたGoppa符号やSavin符号などの代数幾何符号は、代数曲線上のベクトル束の点での評価写像を用いて構成されていたが、当研究ではこれを一般化して高次元射影多様体上の因子への制限写像を用いた一般Savinの概念を導入し、パラメーターの評価などの基本的性質を明らかにした点に学術的意義がある。今後、従来より良いパラメーターを備えた一般Savin符号を構成できれば、情報通信の分野への応用が期待される。