Protograph-Based Generalized LDPC Codes: Enumerators, Design, and Applications

Abstract

Among the recent advances in the area of low-density parity-check (LDPC) codes, protograph-based LDPC codes have the advantages of a simple design procedure and highly structured encoders and decoders. These advantages can also be exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. In this dissertation we provide analytical tools which aid the design of protograph-based LDPC and G-LDPC codes. Specifically, we propose a method for computing the codeword-weight enumerators for finite-length protograph-based G-LDPC code ensembles, and then we consider the asymptotic case when the block-length goes to infinity. These results help the designer identify good ensembles of protograph-based G-LDPC codes in the minimum distance sense (i.e., ensembles which have minimum distances grow linearly with code length). Furthermore, good code ensembles can be characterized by good stopping set, trapping set, or pseudocodeword properties, which assist in the design of G-LDPC codes with low floors. We leverage our method for computing codeword-weight enumerators to compute stopping-set, and pseudocodeword enumerators for the finite-length and the asymptotic ensembles of protograph-based G-LDPC codes. Moreover, we introduce a method for computing trapping set enumerators for finite-length (and asymptotic) protograph-based LDPC code ensembles. Trapping set enumerators for G-LDPC codes represents a more complex problem which we do not consider here. Inspired by our method for computing trapping set enumerators for protograph-based LDPC code ensembles, we developed an algorithm for estimating the trapping set enumerators for a specific LDPC code given its parity-check matrix. We used this algorithm to enumerate trapping sets for several LDPC codes from communication standards. Finally, we study coded-modulation schemes with LDPC codes and pulse position modulation (LDPC-PPM) over the free-space optical channel. We present three different decoding schemes and compare their performances. In addition, we developed a new density evolution tool for use in the design of LDPC codes with good performances over this channel.