Algebraic constructions of nonbinary quasi-cyclic LDPC codes and efficient encoding

This dissertation presents three algebraic methods for constructing nonbinary low-density parity-check (LDPC) codes whose Tanner graphs has girth at least 6. The experimental results show that these codes perform well under iterative decoding algorithm. Compared with other nonbinary LDPC codes, codes constructed by algebraic methods have quasi-cyclic or cyclic structure and therefore allow efficient encoding. First presented is a finite field approach for constructing two classes of quasi-cyclic LDPC codes. The parity-check matrices of the codes constructed by the finite field approach usually have full row rank or nearly full row rank. Hence, the encoding complexity is small. In general, this approach is suitable for constructing high-rate codes, whose parity-check matrices have small column weights. Next a finite geometry approach is presented for constructing one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. The parity-check matrices of the codes constructed by the finite geometry approach usually have large column weights, hence these codes may show a very low error floor. Iterative decoding of these nonbinary LDPC codes converges very fast. Then a superposition-dispersion method is devised for constructing long quasi-cyclic LDPC codes from short codes with small symbol size. The short codes can be constructed by the two previous approaches. Finally, the efficient encoding of quasi-cyclic LDPC codes using shift registers is presented.