Theorem (Kraft Inequality)

A set of positive integers $\{\mathscr{l}_{1},\cdots,\mathscr{l}_{M}\}$ is said to satisfy the Kraft inequality with base $D$ (where $D\ge2$ integer) if $$\sum\limits_{i=1}^{M}D^{-\mathscr{l}_{i}}\le1$$

Theorem (Kraft Inequality for Uniquely Decodably VLCs)

Let $\mathcal{C}$ be a UD $D$-ary n-th order VLC for a discrete source $\{X_i\}_{i=1}^\infty$ with alphabet $\mathcal{X}$ and let $\mathscr{l}_{1},\cdots,\mathscr{l}_{M}$ be the lengths of the code’s $M=|\mathcal{X}|^M$ codewords. Then these codeword lengths satisfy the Kraft Inequality with base $D$ $$\sum\limits_{i=1}^{M}D^{-\mathscr{l}_{i}}\le1$$