This paper considers the problem of finding average enumerators for the class of protograph ensembles, which are related in a certain way to quasi-cyclic codes. Our methods, which are necessarily different from those used to compute enumerators for classical irregular ensembles, can be applied to both codeword and stopping set weight enumerators. The method divides codewords into types based on their partial weight enumerator. For each type, an exponent can be computed for the average number of codewords of that type. Maximizing over types of fixed average weight gives the average enumerator which we seek. Although this maximization step is in general difficult because of non-unique local maxima, we can compute it for simple cases. We show that certain ensembles exist which have a linearly growing minimum distance with high probability, while others have at most sublinearly growing minimum distance with high probability
This paper considers the problem of finding average enumerators for the class of protograph ensembles, which are related in a certain way to quasi-cyclic codes. Our methods, which are necessarily different from those used to compute enumerators for classical irregular ensembles, can be applied to both codeword and stopping set weight enumerators. The method divides codewords into types based on their partial weight enumerator. For each type, an exponent can be computed for the average number of codewords of that type. Maximizing over types of fixed average weight gives the average enumerator which we seek. Although this maximization step is in general difficult because of non-unique local maxima, we can compute it for simple cases. We show that certain ensembles exist which have a linearly growing minimum distance with high probability, while others have at most sublinearly growing minimum distance with high probability
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 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.
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.
In earlier work, we presented a method for finding ensemble weight enumerator for protograph-based generalized LDPC (G-LDPC) codes, and leveraged this method to find ensemble stopping set enumerator and ensemble trapping set enumerator. The method is conceptually simple, but when the dimensionality of the constraint nodes (number of their code-words) grows, it becomes difficult to handle the computational complexity, which rise while evaluating these enumerators. To deal with this difficulty, we posed a conjecture, which greatly reduce the computational complexity. Trails to proof this conjecture showed that the proof is a challenging problem. Also, proving it will strengthen the theory of enumerating protograph-based G-LDPC code ensembles. Which in turn helps in predicating the average performances for codes drawn from these ensembles. In Section II we present a review of our method for finding finite and asymptotic weight enumerators for protograph-based G-LDPC code ensembles. Then, we present the conjecture in Section III with some examples.
Asymptotic analysis of low-density parity-check (LDPC) code weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with linear minimum distance and low error floors. However, the analysis cannot capture the behavior of codewords and stopping sets that grow sublinearly with codelength. Thus, it is unclear how well the analysis describes behavior for finite codelengths, particularly for short codes. In this paper, we provide another perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, protograph structure, and preceding. Using approximations to the stopping-set enumerators, we show that for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly- growing stopping sets. Further, we analyze for what stopping-set sizes and codelengths the approximations apply.
Recently, LDPC codes with projected graph, or protograph structures have been proposed. In this paper, finite length ensemble weight enumerators for LDPC codes with protograph structures are obtained. Asymptotic results are derived as the block size goes to infinity. In particular, we are interested in obtaining ensemble average weight enumerators for protograph LDPC codes which have typical minimum distance that grows linearly with block size. As with irregular ensembles, linear minimum distance property is sensitive to the proportion of degree-2 variable nodes. In this paper, the derived results on ensemble weight enumerators show that linear minimum distance condition on degree distribution of unstructured irregular LDPC codes is a sufficient but not a necessary condition for protograph LDPC codes