# A Low-Complexity Ordered Statistics Decoding Algorithm for Short Polar Codes
## 8 Simulation Results  
### WER Performance  
In this section, we show some simulation results of our proposed algorithm. We tested the WER performance of the proposed decoding algorithm on (64,32) and (128,64) polar codes over a BIAWGN channel and have made other comparisons with the SC, OSD, and SCL decoding algorithm with various list sizes. Figures 3 and 4 plot the WER performance of the decoding algorithms along with the upper-bound computed using Equation (9) with different threshold settings.  
Figure 3: WER performance comparison between SC, OSD, the algorithm proposed in [10] and TH-OSD with codeword length 64 and rate 0.5.  
Figure 2: The structure of hybrid TH-OSD-SC decoding algorithm.  
As the figures show, the OSD algorithm performed the best among all the algorithms with TH-OSD's WER performance below that of OSD. However, as the threshold decreased, TH-OSD's WER performance approached OSD's. The performance of OSD-4 was the same as OSD-2's. These observations are in line with our expectations and confirm that OSD-2 performs similarly to MLD in short-code length scenarios. TH-OSD tested fewer candidate codewords compared to OSD. With a high threshold, TH-OSD performed similarly to SC with low SNR values and outperformed it with mid-to-high SNR values. As the threshold is decreased, TH-OSD's performance improved at the price of testing more codewords. The performance of the proposed algorithm in [10] with threshold \(3\sigma\) is rather poor, and the performance with threshold \(5\sigma\) is close to OSD-2's performance at the price of barely no complexity reduction compared to the original OSD which can be seen in the next subsection. The SC is a suboptimal decoding method since it discards the information provided by further frozen bits when decoding an information bit [4]. As a result, SC's WER performance was not good. SCL with a large list size also achieved performance similar to MLD's, but we excluded its curve from Figure 3 to preserve clarity.  
Figures 3 and 4 also show a noticeable gap between the trajectories of TH-OSD and OSD in low SNR areas. The gap between TH-OSD and OSD was larger with higher threshold settings since a codeword with a higher \(f\) value is less likely to be the transmitted codeword. One way to reduce this gap is to use a list. By using a list, a candidate codeword with an \(f\) value less than the threshold will not be declared immediately as the decoded codeword. Instead, we put the codeword in the candidate list if the list is not full. We terminate the decoding process once the candidate list is full and declare the candidate codeword with the smallest \(f\) value as the decoded codeword. By simulation, we found that a list of size 2 is enough to pull TH-OSD's trajectory towards OSD's. Figure 5 shows the effects of a size 2 list: TH-OSD's WER performance was almost the same as OSD-2's. Figure 5 also shows the performance of CA-OSD and CA-TH-OSD. We simulated length 64 polar code with 4-bit CRC whose generator polynomial is \(g(x)=x^{4}+x+1\). Figure 5 shows the two crc-aided algorithms both outperformed OSD-2. Thus, they also outperformed MLD since OSD-2 performed maximum likelihood decoding in this scenario. CA-TH-OSD performed almost the same as CA-OSD but tested fewer codewords.  
Figure 4: WER performance comparison between SC, OSD, the algorithm proposed in [10] and TH-OSD with codeword length 128 and rate 0.5