  I generalize the concept of balancedness to qudits with arbitrary dimension
$d$. It is an extension of the concept of balancedness in New J. Phys. {\bf
12}, 075025 (2010) [1]. At first, I define maximally entangled states as being
the stochastic states (with local reduced density matrices $\id/d$ for a
$d$-dimensional local Hilbert space) that are not product states and show that
every so-defined maximal genuinely multi-qudit entangled state is balanced.
Furthermore, all irreducibly balanced states are genuinely multi-qudit
entangled and are locally equivalent with respect to $SL(d)$ transformations
(i.e. the local filtering transformations (LFO)) to a maximally entangled
state. In particular the concept given here gives the maximal genuinely
multi-qudit entangled states for general local Hilbert space dimension $d$. All
genuinely multi-qudit entangled states are an element of the partly balanced
$SU(d)$-orbits.
