Q$-3.(20$ pts$)$ Suppose that we have a file consisting of large DNA strings. All the possible characters

are A$,$ C$,$ G$,$ T or blank character $(\_).$

It is known that the most frequent character is A with $40\%$ frequency. The least frequent character is

$\_$ with $10\%$ frequency. We are going to apply Huffman coding.

$($a$-10$ pts$)$ Given only the above information, what is the maximum possible codeword length for

the blank character $"\_"?$ Prove your answer.

$($b $-10$ pts$)$ Apply Huffman coding for the case where the frequencies of $C,G,$ and $T$ are $15\%,17\%,$

and $18\%$ respectively. Find the compression ratio with respect to the minimum fixed$-$length

encoding.