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

本研究涉及相关性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，以便在这些案例中的任何一个方面的进展都将构成戏剧性的进展，并提高我们对整个相关性问题的理解。 这些案例的选择也着眼于实际应用。我们将使用代数方法产生块Hankel加权矩阵，多维Hadamard矩阵，几乎差集和完美序列。我们的动机源于它们在通信工程的几个领域中的实用性：量子计算、MC-CDMA系统、准同步CDMA、多天线无线通信系统、广泛用于军用无线电、CDMA和GSM网络、雷达和声纳以及蓝牙通信的FHSS-仅举几例。可以使用现代代数开发的离散数学结构，数论和有限几何以及其它组合结构在构造具有期望的相关特性的序列和阵列中是有用的。他们通过他们的代数对应系统地研究。所获得的结果将导致新的数学理论，是感兴趣的组合设计理论家和通信工程师。因此，我们调查序列设计问题，这在通信工程中有各种应用。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.