On Convergence Speed of Capacity-Achieving Sequences for Erasure Channel

Oswald and Shokrollahi carried out a systematic study of capacity-achieving sequences of low-density parity-check (LDPC) codes for the binary erasure channel. Two quantities mu and Delta were suggested to measure the convergence speed of these sequences. They gave an upper bound of mu for a certain type of capacity-achieving sequences. In this correspondence, a lower bound of mu for this type of sequences is provided. It follows that, under a certain condition, the exact value of mu can be determined, which answers an open question posed by Oswald and Shokrollahi in their paper. If the condition is not satisfied the upper bound may not be tight. A counterexample is given.