TABLE OF CONTENTS In this chapter, Reed-Muller (RM) codes and Reed-Muller turbo codes are discussed. We present definition and properties of RM codes. It was stated in Chapter 5 that there are two approaches used in MAP decoding of block codes. One is based on the list decoding algorithm, i.e. the modified Chase-II algorithm [128]. It is noted that list decoding is sub-optimal because it does not perform full search over all valid codewords. Trellis-based decoding algorithm, however, is optimal. Our focus will be on the latter. The issue of constructing the trellis of block codes and, particularly their minimal trellis representation, is then considered. Finally, the details of RM-turbo codes, their encoder and decoder are presented. The chapter is organized as follows.
In the second section, the definition and properties of Reed-Muller codes are presented. In the third section, we present the definitions related to the trellis diagram of block codes. Then the construction of the trellis diagram of a linear block code using BCJR [57] and Massey algorithm [145] is discussed. In particular, the construction of trellis diagram of a RM code is presented. Then, turbo encoder and decoder will be presented. The presented encoder is a parallel concatenated code constructed from two elementary encoders with an interleaver between them. The decoder is an iterative MAP decoding algorithm. Then, the system model used for the simulation purpose is given. The simulation results of RM turbo codes on Additive White Gaussian Noise (AWGN) and Rayleigh-fading...
