In this paper, we propose a semi-analytical method to design polar codes for the reconfigurable intelligent surface (RIS)-assisted channel with Rayleigh fading on each wireless link, while also providing an approximate closed-form expression for its word error probability (WEP) as a function of the number of RIS elements. Conventional polar code design methods involve numerically evaluated values of the bit-channel transition probabilities used to obtain the WEP upper bound in which the impact of the number of RIS elements, signal-to-noise ratio (SNR), and code rate is challenging to analyze. In contrast, our approach uses exponential approximations to obtain approximate closed-form expressions for the bit-channel transition probabilities. These exponential terms can be succinctly characterized in terms of coefficient pairs (CPs) which can be tracked numerically during code design and are approximately independent of the SNR and the number of RIS elements. This allows for rapid design of polar codes for any combination of the RIS elements and SNR. Numerical and simulation results show that the proposed method efficiently predicts the required number of RIS elements for a target WEP within the range of$10^{\text {-}2}$and$10^{\text {-}6}$, and can also be used to optimize resource allocation in a multiuser context.

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 × 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.