Constructions of Combinatorial Arrays and Block Designs using Algebraic Curves

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。

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.