Pi01 encodability and omniscient reductions

A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the $\Pi^0_1$ encodable compact sets as those who admit a non-empty $\Sigma^1_1$ subset. Thanks to this equivalence, we prove that weak weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch.