Galois Theory


 

A short tutorial to assist in obtaining an understanding of Reed-Solomon Coding

 

A.A.R.Townsend

29/10/2004

 


Introduction

 

Digital Television Terrestrial Broadcasting (DTTB) uses Coded Orthogonal Frequency-Division Multiplex (COFDM) which is a multi-carrier modulation method in which many closely spaced carriers occupy a given channel bandwidth. The letters FDM corresponds to Frequency-division multiplex and refers to the set of evenly spaced carriers. The letter “C” refers to Coded which in turn refers to the use of channel coding to combat frequency-dependent fading and degradation of the symbol or bit error rate. Finally, “O” refers to Orthogonal which refers to the relationship between the multiple carriers.

What is considered in this tutorial is the “C” or coded component, more specifically, the outer code that is known as the “Reed-Solomon” code.  This outer code is a block code that is effective in coping with large consecutive losses of data, such as might occur with analogue television co-channel interference or impulse noise.  In contrast, the inner code in a COFDM system is a “Viterbi “or trellis code. The trellis code is most effective for coping with random errors such as those caused by white noise. When the trellis code capacity is exceeded, a burst error is generated at the output. It is because of this burst error and also to deal with the occurrence of any impulse noise entering the receiver that the Reed-Solomon outer block decoder is concatenated after de-interleaving and trellis decoding. This permits a more robust system to be obtained. 

This tutorial comprises 278 power point slides that have been converted into pdf format, so that it can be read by adobe reader 6.0 and above and downloaded by anyone wishing to complete the tutorial off-line. It has been called “Galois fields” as it requires a brief introduction to the mathematics of finite fields for a proper understanding of the Reed-Solomon encoding and decoding process. The tutorial can be opened by clicking on “Galois fields

 

Tony Townsend, tonyart@ieee.org