Monday, April 3, 2000
2:15 PM (refreshments 2:00)
RLE Conference Room, Room 36-428
EECS Special Seminar
Abstract
The basic commodity of the scalable information infrastructure of the future will certainly be information represented in discrete alphabet form. Such information representations must be transmitted, stored, and manipulated without error and often with high levels of security. The processing of such discrete data is therefore at the very core of the communications infrastructure. Finite alphabet processing is a key component of error control coding, coding for security, multi-user access, joint source/channel coding, and much more.
Our research at Georgia Tech has shown that processing based on a newly developed ``theory of wavelet transforms over finite fields" promises a framework for new approaches to the fundamental problems mentioned above. This new theory provides a general wavelet decomposition of sequences defined over finite fields --an approach that has a rich history in signal processing for the representation of real-valued signals, but which has been lacking in the finite field case.
The primary focus of this talk is to address wavelets and filter banks over finite fields and their intersection with error correcting codes. We present a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones. First, we study wavelets and filter banks over Galois fields, in particular, paraunitary filter banks over fields of characteristic two . Second, we present the finite field wavelet approach to study error control codes such as block codes including the Golay code and other self-dual codes, and convolutional codes. We have derived wavelet representations of many previously known codes. We have found new codes whose existence has been conjectured but not demonstrated. We have developed wavelet codes that are the best possible codes in that they correct the maximum number of errors for a given amount of redundancy. Finally, we have explored new structures for decoding many types of codes.
We believe that the finite field wavelet transform theory that we have developed has applications that range far beyond error control coding to some of the other areas of communications such as multiple access systems and privacy coding. In all these application areas and many more, the finite field wavelet transform framework allows many opportunities for combining basic error control coding with other processing functions. Such combinations can yield huge dividends in effectiveness and efficiency. We hope that this research will serve as a starting point from which these possibilities might be explored and extended.
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Modified: Apr 3, 2000|
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