The following is a well-know fact:
Theorem The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel set coded in $V$.
Now suppose that $\kappa$ is an uncountable cardinal and let $(r_i: i<\kappa)$ be a sequence of reals.
Question. Is there a characterization theorem as above for the sequence $(r_i: i<\kappa)$ to be $Add(\omega, \kappa)$-generic over $V$, where $Add(\omega, \kappa)$ is the Cohen forcing for adding $\kappa$-many Cohen reals.
If there is a known characterization, giving a reference is welcome.
Remark. I know there are characterizations of Cohen algebra, for example the one given in the paper Characterizations of Cohen algebras, but I'm interested in a characterization parallel to the one given in the above theorem.
Remark 2. The reason I'm asking the question is the following: In the paper Adding many random reals may add many Cohen reals I showed that forcing with $R(\kappa) \times R(\kappa)$ adds a generic filter for $Add(\omega, \kappa)$ (where $R(\kappa)$ is the usual forcing for adding $\kappa$-many random reals), which generalizes the well-known fact that forcing with $R \times R$ adds a Cohen real (where $R$ is the random forcing). The usual proof of the above known result uses characterization of Cohen reals given in the above theorem, while I presented a direct proof without using any characterization result. If there is a characterization as asked above, there might be a different proof parallel to the known one.