Definition (VLC)

Given a discrete source $\{X_i\}^\infty_{i=1}$ with alphabet $\mathcal{X}$ and given a $D$-ary code alphabet $B=\{0,1,\cdots,D-1\}$, $D\ge2$ fixed integer, a $D$-ary $n$-th order variable-length code (VLC) for the source is a map $$f:\mathcal{X}^{n\to}B^*$$mapping source $n$-tuples to $D$-ary codewords of variable lengths, where $$\begin{align*} B^*=\mbox{set of all finite-length strings from }B\\ c\in B^{*}\iff\exists \ l\ge1\mbox{ s.t. }c\in B^l \end{align*}$$The codebook $C$ of the VLC is the set of all codewords $$\mathcal{C}=f(\mathcal{X}^n)=\{f(x^n)\in B^*:x^n\in\mathcal{X}^n\}$$