Consider a discrete memory$-$less source $U$ with symbol alphabet $U=\{u_1,u_2,$dots,$u_8\}.$ The

probabilities of symbols equal $Pr(U=u_i)=p_i,i=1,$cdots,$8.$

$(1)$ Construct an optimal prefix$-$free code for the case $p_1=p_2=0\mathrm{.}03,p_3=0\mathrm{.}04,p_4=0\mathrm{.}1,$

$p_5=0\mathrm{.}15,p_6=0\mathrm{.}2,p_7=0\mathrm{.}2,p_8=0\mathrm{.}25.$ What is the average length of this code? $(3$ points$)$

$(2)$ Construct another optimal prefix$-$free code for this source, not the same as the above. $(1$ point$)$

$(3)$ Construct a non$-$singular code $($all codewords should be different$)$ such that the average length

$L_{av}$ is less than the average length of the prefix$-$free codes above. $(1$ point$)$

Consider now a possibility of arbitrary $p_1,$dots,$p_8.$ For a given code, let us denote $l_{\text{m}\text{i}\text{n}}=mi{n}_i=1,$cdots,$8l_i$

and $l_{\text{m}\text{a}\text{x}}=ma{x}_i=1,$cdots,$8l_i$

$(1)$ Is there a prefix free code with lengths $l_1=l_2=l_3=2,l_4=3,l_5=3,l_6=l_7=l_8=5($explain$,$

construct it is exists$)(2$ points$)$

$(2)$ Is there a prefix free code with lengths $l_1=l_2=2,l_3=l_4=3,l_5=4,l_6=l_7=l_8=5($explain$,$

construct it is exists$)(2$ points$)$

$(3)$ Find $p_1,$dots,$p_8$ and construct an optimal prefix$-$free code such that $l_{\text{m}\text{a}\text{x}}$ is the largest possible.

$(2$ points$)$

$(4)$ Find $p_1,$dots,$p_8$ and construct an optimal prefix$-$free code such that $l_{\text{m}\text{a}\text{x}}$ is the smallest possible.

$(2$ points$)$

$(5)$ Find $p_1,$dots,$p_8$ and construct a non$-$singular code such that $l_{\text{m}\text{a}\text{x}}$ is the smallest possible. $(2$

points$)$

Show your solution.