  Multilevel coding (MLC) is a coded modulation technique which can achieve
excellent performance over a range of communication channels. Polar codes have
been shown to be quite compatible with communication systems using MLC, as the
rate allocation of the component polar codes follows the natural polarization
inherent in polar codes. MLC based techniques have not yet been studied in
systems that use spatial modulation (SM). SM makes the polar code design
difficult as the spatial bits actually select a channel index for transmission.
To solve this problem, we propose a Monte Carlo based evaluation of the ergodic
capacities for the individual bit levels under the capacity rule for a
space-shift keying (SSK) system, where we also make use of a single antenna
activation to approximate the transmission channel for the design of the
multilevel polar code. Our simulation results show that the multilevel polar
coded 16 $\times$ 1 SSK system outperforms the corresponding system that uses
bit-interleaved polar coded modulation by 2.9 dB at a bit-error rate (BER) of
$10^{-4}$.
