CIF: Small: Algebraic Methods in the Study of Some Problems in Communication Engineering

这项研究涉及到对关联问题的一些具体案例的研究，这些案例的实例是经过精心选择的，因此任何这些案例的进步都将构成巨大的进步，并提高我们对整个关联问题的理解。这些案例的选择也着眼于实际应用。我们将使用代数方法来生成块Hankel加权矩阵、多维Hadamard矩阵、几乎差集和完美序列。我们的动机来自于它们在通信工程的几个领域中的应用：量子计算、MC-CDMA系统、准同步码分多址、多天线无线通信系统、广泛应用于军事无线电、CDMA和GSM网络、雷达和声纳以及蓝牙通信的FHSS-举几个例子。离散的数学结构可以用现代代数、数论和有限几何以及其他组合结构来建立，在构造具有理想的相关特性的序列和阵列时是有用的。通过它们的代数对应关系对它们进行系统的研究。所获得的结果将导致组合设计理论家和通信工程师感兴趣的新的数学理论。因此，我们研究在通信工程中有着广泛应用的序列设计问题。我们的方法将是非常代数的，并将使用代数、有限域和代数数论的工具。

This research involves the study of some specific cases of the Correlation Problem, the instances of which are carefully chosen so that advances in any of these cases would constitute dramatic advances, and improve our understanding of the whole Correlation Problem. These cases are also chosen with an eye to practical applications. We would employ algebraic methods to generate block-Hankel weighing matrices, multi dimensional Hadamard matrices, almost difference sets and perfect sequences. Our motivation stems from their usefulness in several areas of communication engineering: quantum computing, MC-CDMA systems , quasi-synchronous CDMA , multiple antenna wireless communication systems , FHSS which are widely used in military radios, CDMA and GSM networks, radars and sonars, and Bluetooth communications - to name a few.Discrete mathematical structures that can be developed using modern algebra, number theory and finite geometry and other combinatorial structures are useful in constructing sequences and arrays with desirable correlation properties. They are systematically studied via their algebraic counterparts. The results obtained will lead to new mathematical theories that are of interest to combinatorial design theorists and communication engineers. We thus investigate sequence design problems, which have a variety of applications in communication engineering. Our methods will be very algebraic and would employ tools from algebra, finite fields, and algebraic number theory.