Abstract

The main focus of this thesis is on finding efficient decoding methods for Reed-Solomon (RS) codes, i.e., algorithms with acceptable performance and affordable complexity. Three classes of decoders are considered including sphere decoding, belief propagation decoding and interpolation-based decoding. Originally proposed for finding the exact solution of least-squares problems, sphere decoding (SD) is used along with the most reliable basis (MRB) to design an efficient soft decoding algorithm for RS codes. For an (N, K ) RS code, given the received vector and the lattice of all possible transmitted vectors, we propose to look for only those lattice points that fall within a sphere centered at the received vector and also are valid codewords. To achieve this goal, we use the fact that RS codes are maximum distance separable (MDS). Therefore, we use sphere decoding in order to find tentative solutions consisting of the K most reliable code symbols that fall inside the sphere. The acceptable values for each of these symbols are selected from an ordered set of most probable transmitted symbols. Based on the MDS property, K code symbols of each tentative solution can he used to find the rest of codeword symbols. If the resulting codeword is within the search radius, it is saved as a candidate transmitted codeword. Since we first find the most reliable code symbols and for each of them we use an ordered set of most probable transmitted symbols, candidate codewords are found quickly resulting in reduced complexity. Considerable coding gains are achieved over the traditional hard decision decoders with moderate increase in complexity. Due to their simplicity and good performance when used for decoding low density parity check (LDPC) codes, iterative decoders based on belief propagation (BP) have also been considered for RS codes. However, the parity check matrix of RS codes is very dense resulting in lots of short cycles in the factor graph and consequently preventing the reliability updates (using BP) from converging to a codeword. In this thesis, we propose two BP based decoding methods. In both of them, a low density extended parity check matrix is used because of its lower number of short cycles. In the first method, the cyclic structure of RS codes is taken into account and BP algorithm is applied on different cyclically shifted versions of received reliabilities, capable of detecting different error patterns. This way, some deterministic errors can be avoided. The second method is based on information correction in BP decoding where all possible values are tested for selected bits with low reliabilities. This way, the chance of BP iterations to converge to a codeword is improved significantly. Compared to the existing iterative methods for RS codes, our proposed methods provide a very good trade-off between the performance and the complexity. We also consider interpolation based decoding of RS codes. We specifically focus on Guruswami-Sudan (GS) interpolation decoding algorithm. Using the algebraic structure of RS codes and bivariate interpolation, the GS method has shown improved error correction capability compared to the traditional hard decision decoders. Based on the GS method, a multivariate interpolation decoding method is proposed for decoding interleaved RS (IRS) codes. Using this method all the RS codewords of the interleaved scheme are decoded simultaneously. In the presence of burst errors, the proposed method has improved correction capability compared to the GS method. This method is applied for decoding IRS codes when used as outer codes in concatenated codes