Non-binary LDPC codes vs. Reed-Solomon codes

This paper investigates the potential of non-binary LDPC codes to replace widely used Reed-Solomon (RS) codes for applications in communication and storage systems for combating mixed types of noise and interferences. The investigation begins with presentation of four algebraic constructions of RS-based non-binary quasi-cyclic (QC)-LDPC codes. Then, the performances of some codes constructed based on the proposed methods with iterative decoding are compared with those of RS codes of the same lengths and rates decoded with the hard-decision Berlekamp-Massey (BM)-algorithm and the algebraic soft-decision Kötter-Vardy (KV)-algorithm over both the AWGN and a Rayleigh fading channels. Comparison shows that the constructed non-binary QC-LDPC codes significantly outperform their corresponding RS codes decoded with either the BM-algorithm or the KV-algorithm. Most impressively, the orders of decoding computational complexity of the constructed non-binary QC-LDPC codes decoded with 5 and 50 iterations of a Fast Fourier Transform based sum-product algorithm are much smaller than those of their corresponding RS codes decoded with the KV-algorithm, while achieve 1.5 to 3 dB coding gains. The comparison shows that well designed non-binary LDPC codes have a great potential to replace RS codes for some applications in communication or storage systems, at least before a very efficient algorithm for decoding RS codes is devised.