## 2 Signal Model
The intersection subspace of user \(i\) and \(j\) can be expressed as follows:  
\[span\Big{(}\mathbf{H}^{[r,i]}\mathbf{v}^{[j,i]}\Big{)}=span\Big{(}\mathbf{H}^ {[r,i]}\mathbf{v}^{[i,j]}\Big{)} \tag{1}\]  
where \(\mathbf{H}^{[r,i]}\) represents the \(N\times M\) dimension channel transmission matrix from user \(i\) to the relay, and \(span(\mathbf{A})\) represents the linear space formed by matrix \(\mathbf{A}\); that is, whether there is a vector \(\mathbf{q}_{i}\in\mathbb{C}^{M\times 1}\) satisfying \(\mathbf{q}=\mathbf{A}_{1}\mathbf{q}_{1}=\mathbf{A}_{2}\mathbf{q}\). This condition can be rewritten as follows:  
\[ \begin{bmatrix}\mathbf{I}_{N}&-\mathbf{A}_{1}&\mathbf{0}\\ \mathbf{I}_{N}&\mathbf{0}&\mathbf{A}_{2}\end{bmatrix}\begin{pmatrix}\mathbf{q }\\ \mathbf{q}_{1}\\ \mathbf{q}_{2}\end{pmatrix} =\mathbf{P}\mathbf{x}=\mathbf{0} \tag{2}\]  
From Formula (2), we can see that the problem of the existence of intersection subspaces is transformed into proving whether there is null space for matrix \(\mathbf{P}\). According to the rank zero theorem of linear algebra, as long as the number of columns of the matrix is greater than the rank of the matrix, then the null space exists. It is easy to see that the size of matrix \(\mathbf{P}\) is \(2N\times(2M+N)\). \[\dim(null(\mathbf{P}))=2M+N-\min\{2N,2M+N\}=2M-N\geq 1 \tag{3}\]  
When the user is equipped with \(M_{i}=K-1\) antennas and the relay is equipped with \(N=K(K-1)/2\) antennas, it can be seen from Equation (3) that there is at least 1-dimensional intersection subspace. In other words, a pair of interactive user information can be aligned in the signal space. Therefore, the sending node designs a precoding vector, so that the following formula is true:  
\[ \begin{array}{l}\mathbf{H}^{[r,1]}\mathbf{v}^{[2,1]}=\mathbf{H}^{[r,2]} \mathbf{v}^{[1,2]}=\mathbf{U}_{12}\\ \mathbf{H}^{[r,1]}\mathbf{v}^{[3,1]}=\mathbf{H}^{[r,3]}\mathbf{v}^{[1,3]}= \mathbf{U}_{13}\\ \vdots\\ \mathbf{H}^{[r,K-1]}\mathbf{v}^{[K,K-1]}=\mathbf{H}^{[r,K]}\mathbf{v}^{[K-1,K] }=\mathbf{U}_{(K-1)K}\end{array}  \tag{4}\]  
After signal space alignment (SSA), the signal received by the relay can be expressed as:  
\[ \begin{array}{l}\mathbf{y}^{[r]}=\sum\limits_{i=1}^{K}\mathbf{H}^{[r,i]} \sum\limits_{j=1,j\
eq i}^{K}\mathbf{v}^{[j,i]}s^{[j,i]}+\mathbf{n}^{[r]}\\ \\ =\left[\begin{array}{cccc}\mathbf{U}_{12}&\mathbf{U}_{13}&\cdots&\mathbf{U}_ {(K-1)K}\end{array}\right]\left[\begin{array}{c}s^{[1,2]}+s^{[2,1]}\\ s^{[1,3]}+s^{[3,1]}\\ \vdots\\ s^{[K-1,K]}+s^{[K,K-1]}\end{array}\right]+\mathbf{n}^{[r]}\\ \\ =\mathbf{U}^{[r]}\mathbf{s}^{[r]}+\mathbf{n}^{[r]}\end{array}  \tag{5}\]where \(s^{[r]}_{ij}=s^{[i,j]}+s^{[j,i]}\quad i,j\in\{1,\ldots,K\},i<j\). The relay adopts zero forcing decoding (ZF) and obtains aligned superimposed signals by inverting matrix \(\mathbf{U}^{[r]}\). The physical layer network coding is used to map and modulate the superimposed signal and is broadcast in BC stage. The relay signals to be transmitted in BC stage are \(s^{[r]}_{12},s^{[r]}_{13},\cdots,s^{[r]}_{(K-1)K}\). For receiving node 1, the desired received signals are the signals \(s^{[r]}_{12},\cdots,s^{[r]}_{1K}\) transmitted by node 2 to node \(K\). \(s^{[r]}_{23},\cdots,s^{[r]}_{(K-1)K}\), form interference for user 1. The null space precoding vector is designed to eliminate unwanted interference signals. \[\mathbf{v}^{[r]}_{ij}\subset\left\{null\left(\mathbf{H}^{[1,r]}\right)\cap \cdots\cap null\left(\mathbf{H}^{[m,r]}\right)\cap\cdots\cap null\left( \mathbf{H}^{[K,r]}\right)\right\},m\
eq i\quad\text{and}\quad m\
eq j \tag{6}\]  
Taking user 1 as an example, the received signal is as follows:  
\[\mathbf{y}^{[1]}=\mathbf{H}^{[1,r]}\sum_{i=1,j>i}^{K}\mathbf{v}^{[r]}_{ij}s^{[ r]}_{ij}+\mathbf{n}^{[1]} \tag{7}\]  
Using ZF decoding, the desired signals \(s^{[r]}_{12},\cdots,s^{[r]}_{1K}\) of user 1 can be obtained, respectively. If BPSK is selected as the modulation mode, the expected signal can be obtained by XOR (\(\oplus\)) operation of the signal sent by the user node itself, and by the received signal.