## 1 Introduction
proposed a cooperative relay transmission only when the performance of the trunk link was good [29]. Then, only a link with enough channel gain ratio over the decision threshold was adopted to deliver the messages [30]. However, this may lead to the loss of good diversity in multi-relay cooperation, and can easily cause too much re-transmission in cooperation due to low transmission efficiency and poor energy utility. Finally, factors such as the instantaneous channel state information CSI, OP, signal-to-noise (SNR) boundary, and bit error rate (BER) should be taken into account in optimization of DF protocols. In the proposed polar coded SDF (PCSDF) scheme, a number of relays are chosen and combined with the optimal CSI compound to optimize the BER performance. According to this conception, efficient polar coded SDF relay selection cooperation is proposed under a cooperative threshold decision derived from the CSI. In addition, the corresponding OP of the proposed relay selection is derived, and the BER with different numbers of relay nodes under the optimal threshold is analyzed. In summary, our main contributions are briefly summed up as follows:  
* **Polar codes with specific rates are designed for fading channels. ** A new construction approach is proposed using polar codes in fading channels. The information bit and frozen indices are reselected based on the channel states. In addition, an optimally designed SNR is used in code design with fixed data rate. Thus, coding gain is obtained and the complexity is reduced by using the improved polar codes. * **Optimal selection is obtained for multi-relay cooperation. ** The scheme provides a candidate relay set to choose the optimally selected relays and sets a cooperative threshold at the destination to decide whether the relay takes part in cooperative communications. Several relays with good channel gain ratios are chosen and combined in order to exceed the decision threshold for joint cooperative reception. In addition, the OP for different numbers of relays are analyzed theoretically using the proposed PCSDF scheme for more accurate analytical expression. * **The influence of distance between the source and the optimal relay is numerically analyzed for better performance. ** The distance between the source and the optimal relay is a crucial factor in determining overall performance. Through numerical analyses, the relationship of the distance between the source and optimal relay is obtained. The link status associated with the proposed relay position obviously affects the cooperative schemes, which can be adopted in practice. The remainder of this paper is organized as follows. In Section 2, a multi-relay cooperation system is introduced. Then, a polar encoding construction based on Rayleigh fading channels and PCSDF cooperation is presented in Section 3. In this section, an optimal relay selection is proposed to improve the BER and OP performance. Subsequently, analytical expressions of the optimal SNR design in polar code construction and the OP of the PCSDF scheme are derived in Section 4. In Section 5, simulation results and numerical analyses are presented to verify the good BER and OP performance of our proposed approach, and the resulting complexity is analyzed concisely. Finally, Section 6 concludes the paper.