Error exponent
In information theory, the error exponent of a channel code or source code over the block length of the code is the logarithm of the error probability. For example, if the probability of error of a decoder drops as e−nα, where n is the block length, the error exponent is α. Many of the information-theoretic theorems are of asymptotic nature, for example, the channel coding theorem states that for any rate less than the channel capacity, the probability of the error of the channel code can be made to go to zero as the block length goes to infinity. In practical situations, there are limitations to the delay of the communication and the block length must be finite. Therefore, it is important to study how the probability of error drops as the block length go to infinity.
The channel coding theorem states that for any ε > 0 and for any rate less than the channel capacity, there is an encoding and decoding scheme that can be used to ensure that the probability of block error is less than ε > 0 for sufficiently long message block X. Also, for any rate greater than the channel capacity, the probability of block error at the receiver goes to one as the block length goes to infinity.
Assuming a channel coding setup as follows: the channel can transmit any of messages, by transmitting the corresponding codeword (which is of length n). Each component in the codebook is drawn i.i.d. according to some probability distribution with probability mass function Q. At the decoding end, ML decoding is done.
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