Definition (Uniquely Decodable Fixed Length Code)

A $(K,n)$ D-ary block code for the source is called uniquely decodable or lossless if the encoder: $$f:\mathcal{X}^n\to\{0,1,\cdots,D-1\}^K$$is an invertible map (i.e. $f$ is bijective) and $g=f^{-1}$ (i.e. the decoder is the inverted map of the encoder).

Definition (Probability of Decoding error)

Given a $(K,n)$ D-ary block code $\mathcal{C}$, with encoder/decoder functions $(f,g)$, the reliability is measured via its probability of decoding error $P_e$: $$P_e:=P(g(f(X^n))\not=X^n)$$ we ideally want this to be as close to zero as possible.

Definition (Uniquely Decodable VLC)

A VLC is uniquely decodable (UD), (or lossless), if all finite sequences of source $n$-tuples are mapped onto distinct sequences of codewords $$(f(x_{1}^{n}),\cdots,f(x_{K}^{n}))=(f(y_{1}^{n}),\cdots,f(y_{K'}^{n})) \ \mbox{ for } K=K' \mbox{ and }x_{i}^{n}=y_{i}^{n}, \ i=1,\cdots,K$$or that f is a injective map on all finite sequences of source $n$-tuples. In other words, any concatenation of codewords can be uniquely decoded.