  In this problem, Alice and Bob, are provided $X_{1}^{n}$ and $X_{2}^{n}$ that
are IID $p_{X_1 X_2}$. Alice and Bob can communicate to Charles over
(noiseless) links of rate $R_1$ and $R_2$, respectively. Their goal is to
enable Charles generate samples $Y^{n}$ such that the triple
$(X_{1}^{n},X_{2}^{n},Y^{n})$ has a PMF that is close, in total variation, to
$\prod p_{X_1 X_2 Y}$. In addition, the three parties may posses shared common
randomness at rate $C$. We address the problem of characterizing the set of
rate triples $(R_1,R_2,C)$ for which the above goal can be accomplished. We
build on our recent findings and propose a new coding scheme based on coset
codes. We analyze its information-theoretic performance and derive a new inner
bound. We identify examples for which the derived inner bound is analytically
proven to contain rate triples that are not achievable via any known
unstructured code based coding techniques. Our findings build on a variant of
soft-covering which generalizes its applicability to the algebraic structured
code ensembles. This adds to the advancement of the use structured codes in
network information theory.
