# Protograph LDPC Code Design for Asynchronous Random Access +
## 4 Numerical Results
### Finite-Length Physical Layer Simulations with the Designed LDPC Codes  
We consider here \((960,480)\) LDPC codes obtained from the base matrices in Section 3.4. The code parameters were chosen to fit in the context of short packet communications. Packets in the order of hundreds of information bits are typical of M2M applications. An example is the early data transmission procedure available in Release 15 of 3 GPP, where terminals are allowed to piggyback data on Message 3 of the PRACH with sizes between 328 and 1000 information bits [2]. Each user selects a codeword uniformly at random from the codebook. The codeword is QPSK modulated, and two instances (replicas) are transmitted. Each of the modulated replicas are sent over the asynchronous RA channel undergoing the protocol rules as described in Section 2. Ideal channel estimation is assumed, so that the receiver can perfectly compensate for the phase offset. After channel estimation, soft-LLR values are computed based on the perfect interference power knowledge. Note that the interference contribution here is the superposition of possibly multiple QPSK-modulated, equal-power signals, with random phase offsets, corresponding to all replicas that are concurrently received. It is important to stress that this assumption departs from the Gaussian interference assumption, especially when the number of interfering replicas is small. A standard belief propagation LDPC decoder with a maximum of 50 decoding iterations is used to counteract the effect of noise and multiple access interference. If the decoder is successful, the replica is removed from the received signal together with its twin, so that no residual interference power is left after cancellation.  
The dashed curves in Figure 9 show the PLR performance versus channel load of the asynchronous RA scheme with replicas protected with a code described by the base matrix \(\mathbf{B_{A}}\) (named ad-hoc PHY), \(\mathbf{B_{5G}^{\pi}}\) (named 5G perm. PHY), or \(\mathbf{B_{5G}}\) (named 5G PHY), respectively. The solid curves are resumed from Figure 8 and depict results for the abstracted physical layer. Observe that the relative gap between the ad-hoc LDPC design, permuted 5G design, as well as the 5G design was preserved, since all three codes suffered from a similar degradation due to finite length effects. For the ad-hoc design, at a target PLR of \(10^{-3}\), the supported channel load was reduced from 0.7 b/s/Hz to \(\sim\) 0.6 b/s/Hz. Similarly, at \(10^{-2}\), the achieved channel load became 0.9 b/s/Hz compared to 1 b/s/Hz when the abstract physical layer was considered. In line with the ad-hoc design, also the 5G code suffered from a degradation of \(\sim\) 0.1 b/s/Hz in the channel load region of interest.  
Interestingly, although the ad-hoc LDPC code was designed for a surrogate channel, both the abstract physical layer and the full physical layer simulations confirmed the beneficial impact on the asynchronous random access protocol. The abstract physical layer model was able to hinge the performance tendencies that were subject to a penalty when one considers the precise physical layer. Such penalty mostly originated from finite-length effects of the channel code. The assumption of Gaussianity for the multiple access interference, as well as constant interfering power appeared to be reasonable and thus can be exploited to simplify the channel code design.