Bounds on distortion due to error for rates below and above the channel capacity

Gallager in 1965 IEEE Trans. Inform. Theory IT-11, 3) gave an elegant proof of coding theorem and obtained an upper bound over the probability of error in terms of reliability function. Its dual has recently been proved by Arimoto [(1973), IEEE Trans. Inform. Theory 19, 357]. We have modified the decoding scheme by bringing in the distortion. This provides an upper bound over distortion due to error when rate distortion function is less than capacity. When rate distortion function is above capacity a lower bound on distortion due to error gives the dual for code words of any block length.