Constructions of Combinatorial Arrays and Block Designs using Algebraic Curves

使用代数曲线构造组合数组和块设计

• 批准号: 11640099
• 负责人: FUJI-HARA Ryoh
• 金额: $ 2.24万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640099/
• 关键词: balanced nested design; balanced array; nested design; code divisible multiple access; optical orthogonal code; resolvable design; (t,m,s)-net; finite affine geometry; Frame; 周波数ホッピング多元接続; 分割型の差詰集合族; Balanced Array; Optical orthogonal code; B

Combinatorial problems like orthogonal arrays, balanced arrays, BIBD, etc, are important themes in mathematics since 18^<th> century. Recently those mathematical problems are great matter of concern in application areas like wireless communication, DNA testing and Finance. In this project, we have aimed to construct those applicable combinatorial arrays and designs using algebraic curves and varieties on finite projective geometries. In 1999, we constructed balanced arrays from algebraic curves using a property of multiplicity. In 2000, we constructed code for optical fiber communications, called optical orthogonal code. For this construction, many algebraic and geometrical properties are used. We also found a construction of balanced arrays using geometrical varieties of quadratic functions over finite field. In 2001, we continued to construct optical orthogonal code. Furthermore, we found some construction of resolvable designs and nested designs.

组合问题，如正交数组、平衡数组、BIBD等，是自18世纪以来数学中的重要主题。近年来，这些数学问题在无线通信、DNA检测和金融等应用领域备受关注。在这个项目中，我们的目标是利用有限射影几何上的代数曲线和变异来构造那些适用的组合阵列和设计。1999年，我们利用多重性构造了代数曲线的平衡数组。2000年，我们构建了用于光纤通信的代码，称为光正交码。对于这种构造，使用了许多代数和几何性质。我们还发现了利用有限域上二次函数的几何变异构造平衡阵列的方法。2001年，我们继续构建光学正交码。此外，我们还发现了一些可解析设计和嵌套设计的构造。

项目成果

R.Fuji-Hara, Y.Miao, J.Wang, J.Yin: "Directed B(K,1;v) with K-{4,5} and {4,6} Related to Deletion/Insertion-Correcting Codes"Journal of Combinatorial Designs. Vol.9. 147-156 (2001)

R.Fuji-Hara、Y.Miao、J.Wang、J.Yin：“与删除/插入校正代码相关的 K-{4,5} 和 {4,6} 的定向 B(K,1;v)

R. Fuji-Hara, Y. Miao, J. Yin: "Optimal (9v, 4, 1) optical orthogonal codes with v a product of primes congruent to 1 modulo 4"SIAM Journal on Discrete Mathematics. Vol 14. 256-266 (2001)

R. Fuji-Hara、Y. Miao、J. Yin：“最优 (9v, 4, 1) 光学正交码，其中 v 是素数的乘积，与 1 模 4 同余”SIAM 离散数学杂志。

R.Fuji-Hara, N.Miyamoto: "Balanced Arrays from Quadratic Functions"Journal of Statistical Planning and Influence. 84. 285-293 (2000)

R.Fuji-Hara、N.Miyamoto：“二次函数的平衡数组”统计规划和影响杂志。

R Fuji-Hara, Y.Miao, S,Kuriki, S.Shinohara: "Balanced nested designs and balanced n-ary designs"Journal of Statistical Planning and Influence. (掲載予定).

R Fuji-Hara、Y.Miao、S、Kuriki、S.Shinohara：“平衡嵌套设计和平衡 n 元设计”统计规划与影响杂志（待出版）。

R.Fuji-Hara,S.Shinohara: "Symmetric Sets of Curves and Combinatorial Arrays"Contemporary Mathematics. 225. 225-230 (1999)

R.Fuji-Hara，S.Shinohara：“对称曲线集和组合数组”当代数学。