Q$3.$ Source Coding: A source alphabet of $4$ symbols has probabilities $p_1>p_2>p_3>p_4>0.$

a$)$ Reason if a prefix$-$free code with lengths $\{1,2,2,3\}$ exists for this source?

b$)$ Consider the following source code $C=\{0,10,11,001\}.$ Is this code $($i$)$ singular, $($ii$)$ uniquely

decodable, $($iii$)$ prefix$-$free?

c$)$ If $p_1=p_3+p_4,$ then show that a Huffman code exists where all codeword lengths are equal.

d$)$ If $p_1=p_3+p_4,$ then show that a Huffman code exists where all codeword lengths are not equal.

e$)$ Show that the average codeword length of the two Huffman codes obtained above in c$)$ and d$)$

are same.