  In the present work we consider the problem of a reconstruction of
eigenfunctions of the Johnson graph $J(n,w)$. We give necessary and sufficient
numerical conditions for a unique reconstruction of an eigenfunction with given
eigenvalue by its values on a sphere of given radius $r$ for $n$ big enough. We
also provide examples of functions equal on the sphere but not equal on the
full vertex set in the case of a failure of these conditions.
