Decoding of expander codes at rates close to capacity

The concatenation of nearly-MDS expander codes of Roth and Skachek, "On Nearly-MDS Expander Codes," Proc. IEEE ISIT\´04, with \´typical\´ LDPC codes is investigated. It is shown that for the rates R = (1 - epsi)C (C is the capacity of the binary symmetric channel (BSC)), under certain condition on the parameters of LDPC codes, these concatenated codes have decoding time linear in their length and polynomial in 1/epsi, and the decoding error probability decays exponentially. These codes are compared to the recently presented codes of Barg and Zemor, "Error Exponents of Expander Codes," IEEE Trans. Inform. Theory, 2002, and "Concatenated Codes: Serial and Parallel," IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have all the aforementioned properties