As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are equivalent or stronger than this axiom are contradictory with $V=L$ and all large cardinal assumptions below it are consistent with $V=L$.

Question: What is the situation for the weaker assumption $V=HOD$ and large cardinal tree? Precisely:

(a) Which one of the known large cardinal axioms (together with $ZFC$) does imply $V\neq HOD$?

(b) Is there any strict border in this case as same as "$0^\sharp$ exists" and "$V=L$"?