This work is a first attempt to analyze and understand the coding gain of full-diversity (MDS) low-density parity-check (LDPC) codes. A diversity population evolution is derived. The steady-state distribution of energy coefficients is found. Based on this new analysis, we show new full-diversity LDPC codes with improved performance. Finally, we describe a family of graph codes where all variable nodes, including parity nodes, achieve full diversity after a large number of decoding iterations. Full-diversity MDS low-density codes are not only useful for digital data transmission over wireless channels, some other applications can be found in situations where coding for a finite number of states is required.