Résumé de la conférence de J. Rosenthal - Journée annuelle 2010

"Convolutional Codes, Mathematical Properties and their Applications"

Convolutional codes are extensively used in many coding applications. In most satellite communication systems and deep space transmissions these codes are part of the overall coding
system. The original design of Turbo Codes also involved convolutional codes. Prior to the arrival of codes on graphs and low density parity check codes convolutional codes were used for their easy handling of soft information appearing on a wireless channel. This was often done in concatenation of some algebraic
code like a Reed Solomon.

It is well known that a convolutional code is essentially a linear system defined over a finite field. Despite this well known connection convolutional codes have been studied in the past mainly by graph theoretic methods and in contrast to the situation of block codes there exist only few algebraic constructions. It is a fundamental problem in coding theory to construct convolutional codes with a designed distance.

A first part of the talk describes the connection between convolutional codes and linear systems.  Using systems theoretic methods we explain how to construct codes with maximal or near maximal free distance. We show how decoding can be viewed as a discrete tracking problem where the received signals have to be
optimally matched with a sequence generated by the encoder. We also report on recent progress in the construction of convolutional codes by algebraic means.

In a second part of the talk we will cover applications of convolutional codes over large alphabets. E.g. such codes can be used for fault tolerant systems or for channels where errors appear as erasures. This has then a direct application for video streaming over packet switched systems like the Internet.