Algebraic Methods in Systems Theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

0072383 Rosenthal Mathematically, 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.