Kraft inequality

The lengths of every decipherable binary code satisfyConversely, any positive integer lengths satisfying this bound can be realized by a prefix code.

The lengths of a finite uniquely decodable code satisfy ${\sum }_a{2}^{-l(a)}\le 1$, even when the code is not a prefix code. Distinct concatenations of $r$ codewords remain distinct; counting the possible binary strings of each total length bounds $({\sum }_a{2}^{-l(a)}{)}^r$ by $r{\max }_{a}l(a)$. Taking $r$th roots proves the claim.

Prescribed positive word lengths admit a decipherable code if and only if they admit a prefix code. Counting length-$L$ concatenations of $m$ words proves the Kraft inequality for any decipherable code; choosing free vertices of the ordered $D$-ary tree constructs a prefix code whenever this inequality holds. Each prefix code has unique decodability.

If a decipherable binary code has $N$ codewords of lengths $l_1,\dots ,l_N$, thenThis follows by applying the entropy lower bound with the uniform source distribution.