Channel Coding: Theory, Algorithms, and Applications

Since there are $k=k_1+\cdots +k_M$ message bits, the rate of the overall multilevel code is given as $R=k/n={\sum }_{i=1}^M{k}_i/n$. The minimum distance of the overall multilevel code is derived as

$d_{\text{min,}\mathcal{C}}={\displaystyle \underset{i\in \{1\text{,}\dots \text{,}M\}}{\min }}{d}_i{\delta }_i^2\text{.}$ (12)

(12)

As an example, consider a multilevel code of block length $n=8$: the first level component code is the repetition code with $k_1=8\text{,}{d}_1$