On Achievable Rates and Complexity of LDPC Codes for Parallel Channels:   Information-Theoretic Bounds and Applications
  The paper presents bounds on the achievable rates and the decoding complexity of low-density parity-check (LDPC) codes. It is assumed that the communication of these codes takes place over statistically independent parallel channels where these channels are memoryless, binary-input and output-symmetric (MBIOS). The bounds are applied to punctured LDPC codes. A diagram concludes our discussion by showing interconnections between the theorems in this paper and some previously reported results. 
Igal Sason and Gil Wiechman