We construct a joint coordination-channel polar coding scheme for strong coordination of actions between two agents $mathsf X$ and $mathsf Y$, which communicate over a discrete memoryless channel (DMC) such that the joint distribution of actions follows a prescribed probability distribution. We show that polar codes are able to achieve our previously established inner bound to the strong noisy coordination capacity region and thus provide a constructive alternative to a random coding proof. Our polar coding scheme also offers a constructive solution to a channel simulation problem where a DMC and shared randomness are together employed to simulate another DMC. In particular, our proposed solution is able to utilize the randomness of the DMC to reduce the amount of local randomness required to generate the sequence of actions at agent $mathsf Y$. By leveraging our earlier random coding results for this problem, we conclude that the proposed joint coordination-channel coding scheme strictly outperforms a separate scheme in terms of achievable communication rate for the same amount of injected randomness into both systems.
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