# Protograph LDPC Code Design for Asynchronous Random Access +
## 3 Code Design for the Asynchronous Random Access Channel
### Code Design for Gaussian Interference
#### 3.4.1 Ad-Hoc LDPC Code Design  
Observe that the interference present in the random access channel may affect the beginning, the end, and sometimes both the beginning and end of a packet. In fact, by analysing the interference patterns, we found that the third case, where the interference affects both the beginning and end of a packet, is a case of limited interest for the code design (although it is not difficult to embed additional constraints in the code design). To see this, consider the collision of two packets. Clearly, the interference here hits the beginning or end of a packet. Consider now that three packets collide such that there exists a packet with interference both at the beginning and at the end. Then, the two remaining packets will experience a lower level of interference, which is either at the beginning or at  
Figure 5: Probability density function of the bit LLRs assuming AWGN and a symbol-synchronous QPSK interferer uniform at random phase (rnd. in the caption), or with a fixed phase offset when \(E_{s}/N_{0}=6\) dB. the end. Therefore, SIC may preferably decode one of those packets and cancel their contribution from the received signal. Similar considerations can be made when four packets collide, and we have a packet with both interference at the beginning and the end. For five or more interfering packets, there could conceivably be cases where a packet with interference at both the beginning and end might be favourable to decode first, but for the code design, we focus on the most frequent cases. Thus, we tailor our code design to these specific interference patterns by targeting codes that are robust w.r.t. interference at the beginning or at the end of a codeword. In the following, we target code designs robust to interference at the beginning or end of a codeword by imposing certain symmetry constraints on the code's base matrix \(\mathbf{B}_{\mathbf{A}}\). Let us split \(\mathbf{B}_{\mathbf{A}}=[\mathbf{B}_{\mathbf{P}}|\mathbf{B}_{\mathbf{T}_{ \mathbf{x}}}]\) into two submatrices, an \(m_{b}\times p_{b}\) matrix \(\mathbf{B}_{\mathbf{P}}\) for the punctured columns and an \(m_{b}\times(n_{b}-p_{b})\) matrix \(\mathbf{B}_{\mathbf{T}_{\mathbf{x}}}\) composed by all columns that are not punctured. For the entries \(b_{i,j}\) of \(\mathbf{B}_{\mathbf{T}_{\mathbf{x}}}\), we impose:  
\[b_{(m_{b}-i-1),(n_{b}-p_{b}-j-1)}\overset{!}{=}b_{i,j} \tag{6}\]  
\(\forall i\in\{0,\ldots,m_{b}-1\}\) and \(\forall j\in\{0,\ldots,n_{b}-p_{b}-1\}\). The symmetry requirement in (6) states that the \(j^{\text{th}}\) column of \(\mathbf{B}_{\mathbf{T}_{\mathbf{x}}}\) from the left shall be equal to the \(j^{\text{th}}\) column from the right, whose elements are however placed in a reversed order. A similar requirement is put on the submatrix \(\mathbf{B}_{\mathbf{P}}\). For the protograph search by means of differential evolution, we fix \(n_{b}\), \(m_{b}\), and \(p_{b}\), such that the code rate \(R_{c}=\frac{n_{b}-m_{b}}{n_{b}-p_{b}}\). We also fix \(\sigma_{n}^{2}\) (e.g., as a result of link budget considerations) and look for protographs that for certain \(\alpha\) allow successful decoding at an interference power \(2\sigma_{t}^{2}\) as high as possible. Thus, the iterative decoding threshold is the maximum interference power for a certain \(\alpha\) and \(\sigma_{n}^{2}\), such that the probability of symbol error in a codeword vanishes. Since \(\alpha\) is the outcome of a random process, we are interested in a code design that is robust for various values of \(\alpha\), denoted by \(\alpha^{(\ell)}\). This implies a multi-target optimization. Due to complexity reasons, we resort to only a few different \(\alpha^{(\ell)}\) for which we simultaneously optimize our base matrices. The Gaussian interference model yields channel LLRs for the two QPSK bit levels that are Gaussian distributed. For this setup, EXIT analysis [40] is known to provide accurate iterative decoding threshold estimates.