# Protograph LDPC Code Design for Asynchronous Random Access +
## 4 Numerical Results
### Abstracted Physical Layer
#### 4.1.3 Simulation Results  
The PLR performance assuming the three different protograph code ensembles from Section 3.4 is shown in Figure 8. For reference, the performance for a capacity achieving code ensemble, one that achieves 97% of capacity and one that achieves 95% of capacity, is also shown (solid black, dot-dashed black and dashed black curves, respectively). In the entire channel load range, the ad-hoc design (base matrix \(\mathbf{B_{A}}\)) visibly outperformed both 5G-based solutions (base matrices \(\mathbf{B_{5G}}\) and \(\mathbf{B_{5G}^{\pi}}\)). The simple yet insightful observation that the beginning and the end of a packet shall be strongly protected by the channel code led to a beneficial performance gain also when the overall multiple access interference comes into play. In particular, for a target PLR of \(10^{-3}\), the channel load supported can be extended from 0.6 b/s/Hz of the 5G-based solutions to 0.7 b/s/Hz of the ad-hoc LDPC design, resulting in a 0.1 b/s/Hz or 17% gain. Similarly, at PLR of \(10^{-2}\), the channel load supported can be extended from 0.9 b/s/Hz of the 5G-based solutions to 1.0 b/s/Hz of the ad-hoc LDPC design, resulting in a 0.1 b/s/Hz or 11% gain. For a fixed channel load operating point, the gain was even more remarkable.  
Finally, observe that there is still a visible gap to capacity achieving code ensembles. One may erroneously conclude that the constant interfering power assumption of the surrogate channel model in Section 3.3 is inaccurate, yielding a design that is penalized on the Gaussian interference channel with varying interference powers. However, we found that even a small loss in error correction performance may have a big effect on the PLR. To underpin this observation, consider a code ensemble that achieves 95% of the outage capacity. Observe that this code ensemble performs similarly to the permuted 5G-like design and worse than our ad-hoc design for the surrogate channel. A possible reason for the drastic impact of the code performance on the asynchronous random access PLR might be due to the SIC: in the case that a replica cannot be correctly decoded due to the sub-optimality of the error correcting code, it may arrest the SIC process and cause multiple packet losses. Consider a code ensemble that achieves 97% of the outage capacity instead of 95%. Observe from the figure that such a minor improvement on the code design can result in gains of up to 0.1 b/s/Hz.  
In order to confirm the trends identified with the abstracted physical layer and to overcome its inherent limitations, in the following Section we will resort to simulate the physical layer performance of the proposed LDPC codes.  
Figure 8: PLR vs. channel load G for the 5G, permuted 5G and ad-hoc LDPC protograph code ensembles for the asynchronous RA setting with an abstracted physical layer. The 5G corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{5G}}\). The 5G perm. corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{5G}}^{\pi}\). Ad-hoc corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{A}}\).