McMillan's Theorem

Every Decipherable code satisfies Kraft’s inequality.

Proof

Let be a decipherable code
with word lengths and
Let

where is the number of ways of choosing codewords with total length .

decipherable
(because any word of length in has at most one preimage under
and there is such words)
Then

Taking we find:

Corollary

A Decipherable code with prescribed word lengths exists
if and only if
there exists a Prefix-Free Code with the same word lengths.