Protograph-based doubly generalized LDPC (DGLDPC) codes are explored in this paper. We extend (and in the process simplify) the technique for computing the ensemble weight enumerators of protograph-based LDPC and GLDPC codes to DGLDPC codes. We find that with careful design, protograph-based DGLDPC codes can have a better asymptotic growth rate of minimum distance than that of the protograph-based LDPC and GLDPC codes. Simulation results confirm that protograph-based DGLDPC codes have a low error floor.
Protograph-based doubly generalized LDPC (DGLDPC) codes are explored in this paper. We extend (and in the process simplify) the technique for computing the ensemble weight enumerators of protograph-based LDPC and GLDPC codes to DGLDPC codes. We find that with careful design, protograph-based DGLDPC codes can have a better asymptotic growth rate of minimum distance than that of the protograph-based LDPC and GLDPC codes. Simulation results confirm that protograph-based DGLDPC codes have a low error floor.
Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. The asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length.
Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators and stopping set enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight and stopping set enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. In the weight enumerator case, the asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. In the stopping set enumerator case, the asymptotic results allows us to determine whether or not the typical smallest stopping set size grows linearly with codeword length.
Protograph-based generalized LDPC (GLDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. Recently, a technique for computing ensemble weight enumerators for GLDPC codes has been published. In the current paper, we investigate the existence of typical minimum distance for protograph-based GLDPC codes. That is, we first upper bound the ensemble weight enumerators for finite-length GLDPC codes based on protographs, and then we consider the sum of weight enumerators. The results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. We provide conditions on the connections of degree-2 variable nodes to constraint nodes (short block codes) to have typical minimum distance. These conditions are related to the minimum distances of the constraint nodes.
For LDPC-like codes such as LDPC, GLDPC, and DGLDPC codes, it is well known that the error floor can be caused by the codewords of small weights or stopping sets of small sizes. In this paper, we investigate the computation of asymptotic weight enumerators such that it becomes a convenient tool to determine a good distribution of code ensembles. In addition, by analyzing the first order approximation, we derive a condition to obtain a negative asymptotic growth rate of the codewords of small linear-sized weights, which is an important constraint for distribution optimization. Also the weight enumerators of turbo and repeat-accumulate codes are investigated. Furthermore, we extend our results to nonbinary DGLDPC codes. Generalization to N-layer and convolutional code based LDPC-like codes are also developed.
Protograph-based LDPC and generalized LDPC (G-LDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. The design of such “protograph-based codes” relies on what is effectively a computer-based search. As such, following Gallager, it is prudent to restrict the search to a “good ensemble,” for example, an ensemble whose minimum distance grows linearly with codeword length. A good ensemble can also mean one with good stopping set, trapping set, or pseudocodeword properties. In this paper, ensemble codeword weight enumerators for finite-length LDPC and G-LDPC codes based on protographs were derived, and then the asymptotic case was considered. The asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length. Then, the codeword weight enumerator technique is adapted to yield ensemble stopping set, trapping set, and pseudocodeword enumerators for protograph LDPC and G-LDPC codes. In this case, the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size, trapping set size, and pseudoweight grows linearly with codeword length. Trapping set enumerators for G-LDPC code ensembles represent a more complex problem which we do not consider here.
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.