The security of adding random noise many time pad
Too tricky for a comment...

Conceptually, your scheme is 110% information theoretic secure, as it's a one time pad($ \downarrow $). But since we tend to use octet based computer architecture these days, $  \mathbb{Z}  $ is more appropriate.

Note 1. Ensure $ |r| \ge |v| $.

Note 2. Don't want to patronise, but the one time pad is thee most contentious and misunderstood aspect of cryptography ever invented. Especially on this forum. Is $r$ truly random or just pseudo-random? It's worrisome that I read 'generate' in a kinda algebraic context...