On the typical minimum distance of protograph-based generalized LDPC codes

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.