Please provide solution for this question.MAC with list codes. A $(2^{n{R}_1},2^{n{R}_2},2^{n{L}_2},n)$ list code for a DM$-$MAC $p(y|x_1,x_2)$

consists of two message sets $1$:$2^{n{R}_1}$ and $1$:$2^{n{R}_2},$ two encoders, and a list de$-$

coder. The message pair $(M_1,M_2)$ is uniformly distributed over :

$\{$:$2^{n{R}_2}].$ Upon receiving $y^n,$ the decoder finds hat$(m{)}_1$ and a list of messages $L_2$sube$[1$:$2^{n{R}_2}]$

of size $|L_2|2^{n{L}_2}.$ An error occurs if hat$(M{)}_1{M}_1$ or if the list $L_2$ does not contain the

transmitted message $M_2,$ i$.$e$.,P_e^{(n)}=P\{\begin{array}{cc}\text{h}\text{a}\text{t}& (M{)}_1{M}_1\end{array}$ or $\{$:$M_2!in{L}_2\}.$ A triple $(R_1,R_2,L_2)$

is said to be achievable if there exists a sequence of $(2^{n{R}_1},2^{n{R}_2},2^{n{L}_2},n)$ list codes

such that $\underset{n}{lim}{P}_e^{(n)}=0.$ The list capacity region of the DM$-$MAC is the closure

of the set of achievable triples $(R_1,R_2,L_2).$

$($a$)$ Find a single$-$letter characterization of the list capacity region of the DM$-$MAC.

$($b$)$ Prove achievability and the converse.