  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.
