Weight distributions: how deviant can you be?

We investigate the asymptotic weight distribution of LDPC (low density parity check) code ensembles. We show that two cases, namely, linear and logarithmic growth of the minimum distance can be distinguished, depending on the degree distribution pair. Furthermore, combining this result with Shokrollahi's (2000) flatness condition, we get the surprising conclusion: if we want to achieve capacity, we cannot have large minimum distance!.