New Algebraic Techniques for Constructing Sequences and Arrays with Good Correlation Properties

构造具有良好相关特性的序列和阵列的新代数技术K.T.Arasu，莱特州立大学数学与统计系，代顿，俄亥俄州45435相关性问题涵盖了一个具有深厚数学内涵的广泛的基本组合问题。相关问题的具体解决方案在通信、实验设计、实验室仪器和制造等领域有着广泛的实际应用。随着技术的变化，需要解决的相关问题的实例也会发生变化。这项研究为解决这个问题开发了新的数学方法。相关问题是设计具有指定维度的序列或阵列，其中的元素从指定的有限集合中选择，使得所有非平凡的周期自相关都位于指定的限制集合中。通常使用二次型来计算自相关。更一般地，人们可以寻找具有良好的自相关和良好的互相关的几个阵列。关联问题分为两个问题：何时存在解决方案，如果存在，解决方案空间是什么样子？我们所研究的序列设计问题在通信工程中有着广泛的应用。所使用的方法将是非常代数的，并将使用代数、有限域、代数数论和表象理论的工具。获得的序列的线性跨度(p-秩)的计算将使用组合工具，结合这里开发的代数工具。

New Algebraic Techniques for Constructing Sequences and Arrays with Good Correlation Properties K.T.ArasuDepartment of Mathematics & Statistics, Wright State University,Dayton, OH 45435 The Correlation Problem covers a broad fundamental combinatorial problem with deep mathematical content. Specific solutions to the Correlation Problem have practical diverse applications in communications, experimental design, laboratory instrumentation and manufacturing. As technology changes so too will the instances of the Correlation Problem which need solving. This research develops new mathematical techniques for attacking the problem. The Correlation Problem is to design sequences or arrays with specified dimensions with entries chosen from a specified finite set so that all non-trivial periodic autocorrelations lie in a prescribed restrictive set. Usually the autocorrelations are computed using a quadratic form. More generally, one may seek several arrays with good autocorrelations and good cross-correlations. The Correlation Problem divides into two questions: When do solutions exist, and if so, what does the solution space look like? The sequence design problems that we investigate have a variety of applications in communication engineering. The methods used will be very algebraic and would employ tools from algebra, finite fields, algebraic number theory and representation theory. Calculation of the linear span (p-ranks) of the obtained sequences will use combinatorial tools, in conjunction with the algebraic tools developed here.