Consider a discrete source of three independent symbols $A,B$ and $C$ with probabilities of occurrence $0\mathrm{.}8,0\mathrm{.}1$ and $0\mathrm{.}1,$ respectively.

$($a$)$ Design a binary Huffman code.

$($b$)$ Determine the average code length, compression ratio and coding efficiency of your code.

Consider an extension alphabet $\{AA,AB,AC,BA,BB,BC,CA,CB,CC\}$ formed by selecting two symbols at a time from the original alphabet. The occurrence probability of two$-$symbolalphabet is obtained by multiplying two individual probabilities, for example, $P(AB)=P(A)*P(B).$

$($c$)$ Design a binary Huffman code with the extension alphabet source.

$($d$)$ Determine the average code length per symbol, compression ratio and coding efficiency of the extension code.

$($Note for a and c$)$

At the initial step of forming a Huffman coding tree, the characters with equal probability of occurrence are listed in the ascending alphabetical order. For example, if characters BB$,$ CB$,$ and BC have the equal probabilities of occurrnce, they must be listed in the order of BB$,$ CB$,$ CC$.$