Near-Optimum Soft Decision Decoding of Non-Binary Linear Codes

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Over the past decade, iterative decoding methods have received a great deal of interest due to the astonishing error performances achieved first by turbo codes, and more recently by low-density parity check (LDPC)codes. The importance of these methods can be best realized by the fast integration of turbo codes in several standards and last year, a binary LDPC code was first selected as a standard. Although this last decisionclearly indicates maturity, several issues remain problematic in the implementation of LDPC codes, especially for moderate lengths, which are required in many communications systems. In fact for these lengths, non binary LDPC codes outperform their binary counterparts, but iterative decoding of non binary LDPC codes does not scale well with the size of the finite field used. Furthermore, most standards still contain error control coding schemes based on Reed-Solomon (RS) codes. Although these codes are very powerful and have been used for decades, there still exists a large gap between the best achievable performance and that achieved in commercial products. This research activities address problems related to both classes of non binary codes and can therefore be divided into two major areas: (1) Near-optimum decoding of Reed-Solomon codes; and (2) Reduced complexity decoding of non binary LDPC codes.Reliability based decoding of RS codes using their binary image has been shown to provide promising results at error rates that can be simulated. However several approaches (especially those based on iterativetechniques) RS codes since they are often designed for very low error rates. This research investigates the development of a new reliability based decoding technique which outperforms all previously proposed onesfor RS codes over GF(256). A tight performance analysis of this new approach for any SNR value is also possible. LDPC codes designed over GF(q) and decoded with the belief propagation (BP) algorithm have beenshown to perform better as q increases, but at the expense of an O(q log(q)) increase in complexity. Furthermore the BP algorithm is often too complex for fast VLSI implementations. For q=2, very efficient reduced complexity versions of the BP algorithm have been proposed with negligible performance degradation. However as q increases, the complexity of these approaches increases in O(q^2) and the performance gap with BP also increases with q. The research involves the development of new reducedcomplexity versions of the BP algorithm over GF(q) which keep all advantages obtained for q=2, but with much lower complexity than existing algorithms.