On the existence of typical minimum distance for protograph-based LDPC codes

In this paper we prove that, for a certain class of protograph-based LDPC codes with degree-two variable nodes, a typical minimum distance exists. We obtain a tight bound on the sum of weight enumerators, up to some weight d*, for the ensemble of finite-length protograph LDPC codes. Then we prove that this sum goes to zero as the block length goes to infinity. Finally, we prove that Pr(d < d*) goes to zero as the block length goes to infinity. This typical minimum distance exists if degree-two nodes have certain connections to the check nodes. This is also important in practice since it identifies a certain class of protograph LDPC codes that have typical minimum distances.