Average Spectra and Minimum Distances of Low-Density Parity-Check Codes over Abelian Groups

Ensembles of regular low-density parity-check codes over any finite Abelian group $G$ are studied. The nonzero entries of the parity matrix are randomly chosen, independently and uniformly, from an arbitrary label group of automorphisms of $G$. Precise combinatorial results are established for the exponential growth rate of their average type-enumerating functions with respect to the code-length $N$. Minimum Bhattacharyya-distance properties are analyzed when such codes are employed over a memoryless $G$-symmetric transmission channel. In particular, minimum distances are shown to grow linearly in $N$ with probability one, and lower bounds are provided for the typical asymptotic normalized minimum distance. Finally, some numerical results are presented, indicating that the choice of the label group strongly affects the value of the typical minimum distance.