  We construct the nodal basis of $C^m$-$P_{k}^{(3)}$ ($k \ge 2^3m+1$) and
$C^m$-$P_{k}^{(4)}$ ($k \ge 2^4m+1$) finite elements on 3D tetrahedral and 4D
simplicial grids, respectively. $C^m$-$P_{k}^{(n)}$ stands for the space of
globally $C^m$ ($m\ge1$) and locally piecewise $n$-dimensional polynomials of
degree $k$ on $n$-dimensional simplicial grids. We prove the uni-solvency and
the $C^m$ continuity of the constructed $C^m$-$P_{k}^{(3)}$ and
$C^m$-$P_{k}^{(4)}$ finite element spaces. A computer code is provided which
generates the index set for the nodal basis of $C^m$-$P_k^{(n)}$ finite
elements on $n$-dimensional simplicial grids.
