Although a great deal of research effort has been expended on the design, construction, encoding, decoding, performance analysis, and applications of binary LDPC codes in communication and storage systems, very little has been done on nonbinary LDPC codes in these respects. The first study of nonbinary LDPC codes was conducted by Davey and MacKay in 1998. In their paper, they generalized the SPA for decoding binary LDPC codes to decode q-ary LDPC codes, called QSPA. Later, in 2000, MacKay and Davey introduced a fast-Fourier-transform (FFT)-based QSPA to reduce the decoding computational complexity of QSPA. This decoding algorithm is referred to as FFT-QSPA. MacKay and Davey's work on FFT-QSPA was further improved by Barnault and Declercq in 2003 and Declercq and Fossorier in 2007. Significant works on the design, construction and analysis of nonbinary LDPC codes didn't appear until the middle of 2000. Results in these works are very encouraging. They show that nonbinary LDPC codes have a great potential to replace widely used RS codes in some applications in communication and storage systems. This chapter is devoted to nonbinary LDPC codes.

Just like binary LDPC codes, nonbinary LDPC codes can be classified into two major categories: (1) random-like nonbinary codes constructed by computer under certain design criteria or rules; and (2) structured nonbinary codes constructed on the basis of algebraic or combinatorial tools, such as finite fields and finite geometries.