Reliability and coding rate
Noise can cause a receiver to misidentify transmitted symbols. Channel coding maps messages into structured codewords, adding redundancy that helps distinguish what was sent. A binary block code that uses n transmitted digits for k message bits has rate k/n. The extra digits consume channel uses rather than creating free capacity.
For a discrete memoryless channel, coding below capacity allows error probability to become arbitrarily small with appropriate sufficiently long codes. Above capacity, such asymptotically reliable transmission is impossible. Arbitrarily small error is not a promise of zero errors in every finite transmission, nor does the theorem alone provide an implementation meeting any desired delay or computational budget.
Repeating each message bit three times illustrates the tradeoff: it gives a rate of one-third, with majority voting as a possible decoder. Repetition can improve reliability when independent flips are sufficiently unlikely, but it is not generally a capacity-achieving design.
Optimizing the input distribution
A discrete channel is described by conditional probabilities P(y|x). In the memoryless model, once the current input is given, the current output does not depend on preceding inputs and outputs. An input distribution P(x) specifies how frequently symbols are used. Mutual information I(X;Y) measures the average reduction in uncertainty about the input obtained by observing the output.
Capacity is C=maxₚ₍ₓ₎ I(X;Y), measured in bits per channel use when logarithms have base two. Maximization characterizes the best permitted use of the channel rather than one arbitrarily chosen message distribution. It changes the signaling statistics, not the channel's noise law. Where power or other input constraints apply, optimization must respect them.
A binary example
In a binary symmetric channel, each input digit is flipped independently with probability p. Its capacity is 1−h₂(p), where h₂(p)=−p log₂p−(1−p)log₂(1−p). Equally likely input digits maximize the mutual information.
At p=0 the capacity is one bit per use. At p=1/2 the output is independent of the input and capacity is zero. At p=1, however, capacity returns to one: every digit is predictably inverted, so the receiver can invert it back. Predictable alteration is not the same as irreducible random uncertainty. A raw flip percentage alone can therefore mislead when interpreting the channel.
Bandwidth is not capacity
For an ideal band-limited additive white Gaussian noise channel with bandwidth B hertz, signal power limited to S and in-band noise power N, capacity is C=B log₂(1+S/N) bits per second. Both bandwidth and signal-to-noise ratio enter the expression. Hertz and bits per second measure different things.
For example, B=1000 hertz and S/N=3 give a theoretical capacity of 2000 bits per second under this model. This is not a guarantee for every link occupying that bandwidth. Finite codes, protocol overhead and implementation limits affect achieved throughput. Comparing a measured speed with a theoretical capacity requires matching the assumptions, resource constraints and units rather than equating an advertised bandwidth with usable information rate.