Algebraic Methods in Systems Theory

从数学上讲，卷积码可以看作是有限域上的线性系统。对这些码的研究需要对线性系统的代数表示有很好的理解。因此，拟议项目涉及数学系统理论和编码理论中的若干问题。该方案的主要目的是：1.构造自由距离大、阶数相对较小的卷积码的新方法。2.以代数方式解决卷积码译码问题的新技术。3.研究了构成低密度奇偶校验码的自然推广的一类低密度奇偶校验码。在这个项目中，最终的目标是用代数方法来构造低密度奇偶校验卷积码，它的编、译码复杂度接近线性，性能接近容量。卷积码被用于许多通信系统的数据传输中。应用范围从机载卫星传输系统到地面电话线。从深空传输的大多数图像都以这样或那样的方式使用卷积码进行编码。构造新的高效的卷积码是本研究的目标。拥有这样的新代码将有几个好处。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，这项数学研究还可能导致一种新的密码协议。

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.