Theory of algebraic curves with application toward the coding theory

代数曲线理论及其在编码理论中的应用

• 批准号: 13640048
• 负责人: HOMMA Masaaki
• 金额: $ 1.34万
• 依托单位: Kanagawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640048/
• 关键词: Algebraic Curve; Special Divisor; Weierstrass pair; AG Code; Error correcting Code; Hermitian Curve; 誤り訂正符号; 代数幾何符号; 特殊線形系; Castelnuovo's bound; 国際研究者交流; ブラジル:韓国

The initial purpose of this research was to study algebraic curves with distinctive properties and apply its fruit to coding theory. We list the very short summaries of our harvests that the chief of the project was mainly concerned with.1. We proposed the new invariant "the order of speciality" of a special divisor while we were studing the classical theory of Castelnuovo, and gave a classification of the curves that are extremal in the sense of the invariant.2. We considered the two points AG-code and confirmed that its parameters are better than those of a one point code in a certain situation. Moreover we started to investigate all two points codes on a Hermitian curve, however it is still in progress.3. We organized the international conference "Curve 02" at Yamaguchi university January 28 - 31, 2002, and then edited the proceedings.

这项研究的最初目的是研究具有独特性质的代数曲线并将其成果应用于编码理论。我们列出了项目负责人主要关心的收获的非常简短的总结。1。我们在研究经典Castelnuovo理论时，提出了特殊除数的新不变量“特殊阶”，并对不变量意义上的极值曲线进行了分类。 2．我们考虑了两点 AG 代码，并确认在某种情况下其参数优于单点代码。此外，我们开始研究埃尔米特曲线上的所有两点代码，但仍在进行中。 3．我们于2002年1月28日至31日在山口大学组织了国际会议“Curve 02”，并编辑了会议记录。

项目成果

Kim, S.J., Komeda, J.: "Weierstrass semigroups of a pair of points whose first non-gaps are three"Geom. Dedicata. 93. 113-119 (2002)

Kim, S.J., Komeda, J.：“一对点的 Weierstrass 半群，其第一个非间隙是 3”Geom。

T.Kato.: "Martens' dimension theorem for curves of odd gonality"J. Korean Mats Soc.. 39. 665-680 (2002)

T.Kato.：“奇角性曲线的马滕斯维数定理”J.

T.Kato.: "A characterization of the Fermat curve"Nonlinear Analysis. 47. 5479-5489 (2001)

T.Kato.：“费马曲线的表征”非线性分析。

T.Kato., and Miyako Yamada.: "Projective systems whose supports consist of the union of three linear subspaces"Bull. Korean Math. Soc.. 38. 689-699 (2001)

T.Kato. 和 Miyako Yamada.：“其支撑由三个线性子空间的并集组成的投影系统”Bull。

S.J.Kim, J.Komeda: "Numerical semigroups which cannot be realized as semigroups of Galois Weierstrass semigroups"Arch. Math.. 76. 265-273 (2001)

S.J.Kim，J.Komeda：“不能实现为 Galois Weierstrass 半群的半群的数值半群”Arch。