| Low density parity check (LDPC) codes, discovered by Gallager in 1962, were rediscovered in the late 1990's and shown to form a class of Shannon limit approaching codes in the early 2000's. Ever since their rediscovery, design, construction, analysis and applications of these codes have become focal points of research. This dissertation presents several algebraic, combinatorial and graphical methods for constructing structured regular and irregular LDPC codes systematically. Construction methods presented include: (1) construction of LDPC codes based on finite fields and construction of LDPC codes combined with masking technique; (2) construction of quasi-cyclic (QC)-LDPC codes based on parity check matrices of Reed-Solomon (RS) codes; (3) construction of QC-LDPC codes from three classes of balanced incomplete block designs; (4) a trellis-based algorithm for removing short cycles from bipartite graphs and its application for constructing LDPC codes; (5) construction of LDPC codes for burst-erasure channels. We particularly emphasize the construction of QC-LDPC codes that can be efficiently encoded with simple shift-registers. These codes perform very well over the AWGN and binary random and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor and rate of decoding convergence, collectively. |
