Tighter Decoding Reliability Bound for Gallager's Error-Correcting Code
  Statistical physics is employed to evaluate the performance of error-correcting codes in the case of finite message length for an ensemble of Gallager's error correcting codes. We follow Gallager's approach of upper-bounding the average decoding error rate, but invoke the replica method to reproduce the tightest general bound to date, and to improve on the most accurate zero-error noise level threshold reported in the literature. The relation between the methods used and those presented in the information theory literature are explored. 
Yoshiyuki Kabashima, Naoya Sazuka, Kazutaka Nakamura and David Saad