$2$ Straightforward

$[70]$

Let $x=\{a$:$0\mathrm{.}80,b$:$0\mathrm{.}10,c$:$0\mathrm{.}05,d$:$0\mathrm{.}03,e$:$0\mathrm{.}017,f$:$0\mathrm{.}003\}$ be an alphabet of $6$ symbols.

$2+3+(1+1)+(2+3)+8=20$

$($a$)$ Compute the entropy $H(x).$

$($b$)$ Design a Huffman Code $C$:$x\{0,1{\}}^{+}$for $x.$

$($c$)$ Compute the average code length $L(C)$ and the efficiency $=\frac{H(x)}{L(C)}$ of your coding.

$($d$)$ Encode the string $x=$ baaaacbadaafaaae over $x$ by your $C.$ Show that your $C(x)$ can be uniquely decoded to $x.$

$($e$)$ Design an FSM to decode $C.$