Algebraic Methods in Systems Theory

系统论中的代数方法

• 批准号: 0072383
• 负责人: Joachim Rosenthal
• 金额: $ 11.9万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072383&HistoricalAwards=false
• 关键词: Algebraic; Methods; Systems; Theory

0072383RosenthalMathematically, convolutional codes can be viewed as linear systems over a finite field. The study of these codes requires a good understanding of the algebraic representation of linear systems. The proposed project therefore addresses a number of issues in mathematical systems theory and in coding theory. The main objectives of the proposal are: 1. New methods for constructing convolutional codes with a large free distance and a relatively small degree. 2. New techniques to tackle the decoding problem of convolutional codes in an algebraic manner. 3. An investigation of the class of low density parity check convolutional codes which constitutes a natural generalization of the class of low density parity check codes. For this project it is the ultimate goal to algebraically construct low density parity check convolutional codes whose encoding and decoding complexity is `near linear' and whose performance is `near capacity'.Convolutional codes are used in the data transmission of many communication systems. Applications range from airborne satellite transmission systems to terrestrial telephone lines. Most pictures transmitted from deep space involve in one way or another some encoding with a convolutional code. It is the goal of the proposed research to construct new powerful convolutional codes which can be efficiently encoded and decoded. Having such new codes would have several benefits. First and for all it would allow the construction of smaller and more energy efficient transmission devices which are still capable of doing reliable data communication. The research project will necessitate a mathematical investigation into the algebraic structure of convolutional codes. As it was outlined in the proposal this mathematical research could also lead to a new cryptographic protocol.

从数学上讲，卷积码可以看作是有限域上的线性系统。研究这些代码需要对线性系统的代数表示有很好的理解。因此，提议的项目解决了数学系统理论和编码理论中的一些问题。该提案的主要目标是：1。构造自由距离大、自由度小的卷积码的新方法。2. 以代数方式解决卷积码解码问题的新技术。3. 研究了低密度奇偶校验卷积码，它是低密度奇偶校验码的自然推广。对于这个项目，它的最终目标是代数构造低密度奇偶校验卷积码，其编码和解码的复杂性是“接近线性”，其性能是“接近容量”。卷积码用于许多通信系统的数据传输。应用范围从机载卫星传输系统到地面电话线路。大多数从深空传输的图片都以某种方式使用卷积码进行编码。本文的研究目标是构建新的强大的卷积码，使其能够有效地编码和解码。拥有这样的新法规将有几个好处。首先，它将允许建造更小、更节能的传输设备，这些设备仍然能够进行可靠的数据通信。该研究项目将需要对卷积码的代数结构进行数学研究。正如提案中概述的那样，这项数学研究也可能导致新的加密协议。