On the typical minimum distance of protograph-based non-binary LDPC codes

This paper proves the existence of the typical minimum distance for certain ensembles of the protograph-based nonbinary (PB NB) LDPC codes with degree-2 variable nodes. Using recently obtained ensemble weight enumerators for the PB NB LDPC codes, we show that asymptotically in the code length N, the entries in the weight enumerators up to some weight d* = δ* N (δ* >; 0) vanish under certain conditions on degree-2 nodes. As a consequence, the probability that the code minimum distance is less than d* goes to zero as the code length goes to infinity. Results of this type advance the understanding of structured non-binary LDPC codes.