$!$ A memoryless source emits messages $m_1$ and $m_2$ with probabilities $0\mathrm{.}8$ and $0\mathrm{.}2,$ respectively.

Find the optimum $($Huffman$)$ binary code for this source as well as for its second$-$ and

third$-$order extensions $($i$.$e$.,$ for $N=2$ and $3).$ Determine the code efficiencies in each case.

The Huffman code for the source is simply $0$ and $1,$ giving $L=1,$ and

$H(m)=-(0\mathrm{.}8log0\mathrm{.}8+0\mathrm{.}2log0\mathrm{.}2)$

$=0\mathrm{.}72$ bit