Synthesizing Correlated Randomness using Algebraic Structured Codes

In this problem, Alice and Bob, are provided
$X_{1}^{n}$
and
$X_{2}^{n}$
that are IID
$px_{1}x_{2}$
. 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
$\prod p_{X_{1}X_{2}\mathrm{Y}}$
. 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.