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.

我们研究了hermitian曲线上的两点代码y^q+y = x^<q^+>在q^2元素的场f上，其中q^2是素数的幂。作为这些代码的两个点，我们可以选择无穷大p和方程相对于原点q处的点。我们用C（m，n）表示由线性系统L（MP+NQ）产生的代码。第一个结果是，要考虑小于或等于】n的范围的两点代码C（m，n）小于或等于】q是足够的。在该研究项目的第一年中，我们成功地确定了该范围内所有（m，n）的C（m，n）维度，并找到n = 0和q的最小距离。在第二年，我们很高兴成功地找到了所有n的最小距离C（m，n），其0【少于或等于】n【小于或等于或等于】q。此外，作为第三个结果的一个colrollary，我们在上一篇论文中找到了与ω-构造的两点代码的示例（与S.J kim，goppa appl appl appl appl appl appl appl appl appl appl appl appl appl appl pure weiers pure weierspra​​ pure weierstrass pears pure，在上一篇论文中解释的两点代码的最小距离的估计。

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