In information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays exponentially with the block length of the code. Formally, it is defined as the limiting ratio of the negative logarithm of the error probability to the block length of the code for large block lengths. For example, if the probability of error ${\displaystyle P_{\mathrm {error} }}$ of a decoder drops as ${\displaystyle e^{-n\alpha }}$, where ${\displaystyle n}$ is the block length, the error exponent is ${\displaystyle \alpha }$. In this example, ${\displaystyle {\frac {-\ln P_{\mathrm {error} }}{n}}}$ approaches ${\displaystyle \alpha }$ for large ${\displaystyle n}$. 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.

Classical channel-coding exponents include achievability bounds, such as random-coding exponents, and converse bounds. For discrete memoryless channels, Suguru Arimoto gave a converse bound showing exponential decay of the correct decoding probability for rates above capacity.

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 ${\displaystyle M=2^{nR}\;}$ 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, maximum likelihood decoding is done.

Let ${\displaystyle X_{i}^{n}}$ be the ${\displaystyle i}$th random codeword in the codebook, where ${\displaystyle i}$ goes from ${\displaystyle 1}$ to ${\displaystyle M}$. Suppose the first message is selected, so codeword ${\displaystyle X_{1}^{n}}$ is transmitted. Given that ${\displaystyle y_{1}^{n}}$ is received, the probability that the codeword is incorrectly detected as ${\displaystyle X_{2}^{n}}$ is:

The function ${\displaystyle 1(p(y_{1}^{n}\mid x_{2}^{n})>p(y_{1}^{n}\mid x_{1}^{n}))}$ has upper bound

for ${\displaystyle s>0\;}$ Thus,

Since there are a total of M messages, and the entries in the codebook are i.i.d., the probability that ${\displaystyle X_{1}^{n}}$ is confused with any other message is ${\displaystyle M}$ times the above expression. Using the union bound, the probability of confusing ${\displaystyle X_{1}^{n}}$ with any message is bounded by: