Asymptotic ensemble enumerators for protograph-based generalized LDPC codes: Computational complexity

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.