Error correcting codes from the viewpoints of algebraic curves and finite geometry

从代数曲线和有限几何的角度看纠错码

• 批准号: 15500017
• 负责人: HOMMA Masaaki
• 金额: $ 1.28万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15500017/
• 关键词: Error correcting code; A Gcode; Hermitian Curve; Two-point code; Minimum distance; Positive Characteristic; 国際情報交換; 大韓民国; 設計距離

We studied two-point codes on the Hermitian curve y^q+y=x^<q^+> over the field F of q^2 elements, where q^2 is a power of a prime number. As the two points of those codes, we may choose the point at infinity P and the origin Q with respect to the equation. We denote by C(m, n) the code arising from the linear system L(mP+nQ).Our problems were to compute dim C(m, n) and to find the minimum distance of C(m, n). First result is that it is enough to consider the two-point codes C(m, n) for the range 0【less than or equal】n【less than or equal】q. In the first year of this research project, we succeeded in determining the dimension of C(m, n) for all (m, n) in this range and finding the minimum distance for n=0 and q. In the second year, we happily succeeded in finding the minimum distance C(m, n) for all n with 0【less than or equal】n【less than or equal】q.Moreover, as a corollary of the third result, we found the example of two-point code with Ω-construction in our previous paper (with S.J Kim, Goppa codes with Weierstrass pairs, Pure Appl.Algebra 162(2001)) showed the sharpness of the estimation of the minimum distance of a two-point code that explained in the previous paper.

我们在遗传曲线上研究了两个点代码y^q+y = x^<q^+>在这些代码的两个点上，我们可以选择P和等式的原点Q。 （m，n）由线性系统（MP+NQ）产生的代码。您的问题是计算DIM C（M，N），并找到C的最小距离（M，N）。足以满足范围0 [小于或相等] n [小于或相等]的c（m，n）q。在n = 0和q的最小距离中。在第二年，我们高兴地，所有n的最小距离C（m，n），为0（小于或相等），作为第三结果的推论，是在我们的代数162（2001）中，具有ω构成的两点代码显示了上一篇论文中解释的两点代码最小距离的估计的清晰度。

项目成果

本間正明: "Conics with a Hermitian curve"Symposium on Algebraic Geometry at Niigata,2004報告集. To appear.

本间正明：《带有埃尔米特曲线的圆锥曲线》新泻代数几何研讨会报告集，出版。

Hermitian曲線上の$2$点符号(予報)

埃尔米特曲线上的 $2$ 点符号（预测）

• 发表时间: 2004
• 期刊: 数理解析研究所講究録 1361
• 作者: 本間正明;本間正明
• 通讯作者: 本間正明

The Two-Point Codes on a Hermitian Curve with the Designed Minimum Distance

• DOI: 10.1007/s10623-004-5661-x
• 发表时间: 2006
• 期刊: Designs, Codes and Cryptography
• 作者: M. Homma;S. Kim

本間正明: "Hermitian曲線上の2点符号(予報)"数理研講究録(符号と暗号の代数的数理). To appear.

本间正明：《埃尔米特曲线上的两点码（预测）》数学研究讲座记录（代码与密码学的代数数学）出现。

Toward the Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

• DOI: 10.1007/s10623-004-3807-5
• 发表时间: 2005-10
• 期刊: Designs, Codes and Cryptography
• 作者: M. Homma;S. Kim