Channel Capacity Under Exponentially Decreasing Error Probability

We study the capacity of a general channel under the requirement that the error probability decreases exponentially to zero. A formula of this capacity is established by using information spectrum methods. Specifically, we show that this capacity equals to the supremum, over all input processes, of the input-output inf-information rate under exponential decreasing constraint. We also show that, when the code rate is below the capacity, the probability for a random code not having exponentially decreasing error probability is double exponentially small.