Algebraic constructions of high performance and efficiently encodable non-binary quasi-cyclic LDPC codes

This dissertation presents a general method and eight algebraic methods for constructing high performance and efficiently encodable non-binary quasi-cyclic LDPC codes based on arrays of special circulant permutation matrices. Two design techniques, array masking and array dispersion, for constructing both regular and irregular LDPC codes with desired specifications are also proposed. Codes constructed based on these methods perform very well over the AWGN channel and flat fading channels. With iterative decoding using a Fast Fourier Transform based sum-product algorithm, they achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm. Also presented is a class of asymptotically optimal LDPC codes for correcting bursts of erasures. Due to their quasi-cyclic structure, these non-binary LDPC codes can be encoded using simple shift-registers with linear complexity. Structured non-binary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication or storage systems for combating mixed types of noise and interferences.