Let be an alphabet. A code over is any subset of the set of words on the alphabet such that has “uniquue factorization into letters,” i.e., such that for whenever , with all , then we have and for all . In other words, every “word over ” generated by (considered as an alphabet) can be uniquely factored into “letters” in C.

An example of a subset of which is not a code is given by . Here the word can be written either as or as in terms of elements of . Since nor , is not a code.

If we fix a length for the words, i.e. we require that , then we call a block code, and call the block length of the code. An important property of a code is the code's minimum distance, given by the minimum Hamming distance between any pair of words in .

This notion of code is obviously very general. In practice (i.e., in coding theory) one typically takes codes with a little more structure. See, in particular, linear codes.