Non-binary compound codes based on single parity-check codes.
Shannon showed that the codes with random-like codeword weight distribution are capable of approaching the channel capacity. However, the random-like property can be achieved only in codes with long-length codewords. On the other hand, the decoding complexity for a random-like codeword increases exponentially with its length. Therefore, code designers are combining shorter and simpler codes in a pseudorandom manner to form longer and more powerful codewords. In this research, a method for designing non-binary compound codes with moderate to high coding rate is proposed. Based on this method, non-binary single parity-check (SPC) codes are considered as component codes and different iterative decoding algorithms for decoding the constructed compound codes are proposed. The soft-input soft-output component decoders, which are employed for the iterative decoding algorithms, are constructed from optimal and sub-optimal a posteriori probability (APP) decoders. However, for non-binary codes, implementing an optimal APP decoder requires a large amount of memory. In order to reduce the memory requirement of the APP decoding algorithm, in the first part of this research, a modified form of the APP decoding algorithm is presented. The amount of memory requirement of this proposed algorithm is significantly less than that of the standard APP decoder. Therefore, the proposed algorithm becomes more practical for decoding non-binary block codes. The compound codes that are proposed in this research are constructed from combination of non-binary SPC codes. Therefore, as part of this research, the construction and decoding of the non-binary SPC codes, when SPC codes are defined over a finite ring of order q, are presented. The concept of finite rings is more general and it thus includes non-binary SPC codes defined over finite fields. Thereafter, based on production of non-binary SPC codes, a class of non-binary compound codes is proposed that is efficient for controlling both random-error and burst-error patterns and can be used for applications where high coding rate schemes are required. Simulation results show that the performance of the proposed codes is good. Furthermore, the performance of the compound code improves over larger rings. The analytical performance bounds and the minimum distance properties of these product codes are studied.