One of the most important technologies used in modern communication systems is channel coding. Channel coding dates back to a paper published by Shannon in 1948  entitled “A Mathematical Theory of Communication”. The basic idea behind channel coding is to send redundant information (parity) together with a message to make the transmission more error resistant. There are different types of codes that can be used to generate the parity required, including block, convolutional and concatenated codes. A special subclass of codes consisting of the codes mentioned in the previous paragraph, is sparse graph codes. The structure of sparse graph codes can be depicted via a graphical representation: the factor graph which has sparse connections between its elements. Codes belonging to this subclass include Low-Density-Parity-Check (LDPC) codes, Repeat Accumulate (RA), Turbo and fountain codes. These codes can be decoded by using the belief propagation algorithm, an iterative algorithm where probabilistic information is passed to the nodes of the graph. This dissertation focuses on noisy decoding of fountain codes using belief propagation decoding. Fountain codes were originally developed for erasure channels, but since any factor graph can be decoded using belief propagation, noisy decoding of fountain codes can easily be accomplished. Three fountain codes namely Tornado, Luby Transform (LT) and Raptor codes were investigated during this dissertation. The following results were obtained: <ol> <li>The Tornado graph structure is unsuitable for noisy decoding since the code structure protects the first layer of parity instead of the original message bits (a Tornado graph consists of more than one layer).</li> <li> The successful decoding of systematic LT codes were verified.</li> <li>A systematic Raptor code was introduced and successfully decoded. The simulation results show that the Raptor graph structure can improve on its constituent codes (a Raptor code consists of more than one code).</li></ol> Lastly an LT code was used to replace the convolutional incremental redundancy scheme used by the 2G mobile standard Enhanced Data Rates for GSM Evolution (EDGE). The results show that a fountain incremental redundancy scheme outperforms a convolutional approach if the frame lengths are long enough. For the EDGE platform the results also showed that the fountain incremental redundancy scheme outperforms the convolutional approach after the second transmission is received. Although EDGE is an older technology, it still remains a good platform for testing different incremental redundancy schemes, since it was one of the first platforms to use incremental redundancy.
Dissertation (MEng)--University of Pretoria, 2009.