Generalizing a technique given by Wolfowitz, we calculate a formula for the asymptotic value of the error exponent with random coding on a discrete memoryless channel. We then evaluate this formula analytically for low rates and show that the exact value given by this formula agrees with the random coding exponent of Gallager and Fano. This proves that their exponent, which was only known to be a bound for low rates, gives the exact value and this brings out the inherent limitation of the random coding method.
