A code is a representation of data, represented as an map between a set and the Kleene star of another set. These sets are often called the Source alphabet, and the code alphabet, respectively.

Most codes that we consider are nonsingular, for which the map is injective

The extension $C^*$ of a code $C$ is the mapping from finite length strings of the source alphabet to finite-length strings of the code alphabet defined by $C(x_1,x_2,...,x_n) = C(x_1)C(x_2)...C(x_n)$

A code is called Uniquely decodable if its extension is nonsingular.

We also have Prefix code, which are as efficient as general uniquely decodable codes. Kraft-McMillan theorem, says that we can find a prefix code with the same code lengths as any uniquely decodable code. Can prove using a tree construction (see Kraft inequality), Probabilistic method, or others..

If the source alphabet itself is made of strings of a fixed length from another alphabet, these are called blocks, and the code is called a block code.

Otherwise we have a symbol code.