MAC with a helper. Consider the $3-$sender DM$-$MAC $p(y|x_1,x_2,x_3)$ with mes$-$

sage pair $(M_1,M_2)$ uniformly distributed over $[1$:$2^{n{R}_1}][1$:$2^{n{R}_2}]$ as depicted in

Figure $5\mathrm{.}12.$ Sender $1$ encodes $M_1,$ sender $2$ encodes $M_2,$ and sender $3$ encodes

$(M_1,M_2).$ The receiver wishes to recover both messages. Show that the capacity

region is the set of rate pairs $(R_1,R_2)$ such that

$R_{1}I(x_1,x_3$;$Y|x_2,Q),$

$R_{2}I(x_2,x_3$;$Y|x_1,Q),$

$R_1+R_{2}I(x_1,x_2,x_3$;$Y|Q)$

for some $pmfp(q)p(x_1|q)p(x_2|q)p(x_3|x_1,x_2,q)$ with $|Q|2.$

Remark: This is a special case of the capacity region of the general $l-$message $k-$

sender DM$-$MAC established by Han $(1979).$

Figure $5\mathrm{.}12.$ Three$-$sender DM$-$MAC with two messages.