## 2 Three-Dimensional Single Weight Zero Cross Correlation Code Construction
\begin{bmatrix}\mathbf{1}&0&0&\mathbf{0}&0&0&\mathbf{0}&0&0\\ \mathbf{0}&0&0&\mathbf{0}&0&0&\mathbf{1}&0&0\\ \mathbf{0}&0&0&\mathbf{1}&0&0&\mathbf{0}&0&0\\ 0&\mathbf{0}&0&0&\mathbf{1}&0&0&\mathbf{0}&0\\ 0&\mathbf{1}&0&0&\mathbf{0}&0&0&\mathbf{0}&0\\ 0&\mathbf{0}&0&0&\mathbf{0}&0&0&\mathbf{1}&0\\ 0&0&\mathbf{0}&0&0&\mathbf{0}&0&0&\mathbf{1}\\ 0&0&\mathbf{0}&0&0&\mathbf{1}&0&0&\mathbf{0}\\ 0&0&\mathbf{1}&0&0&\mathbf{0}&0&0&\mathbf{0}\end{bmatrix}  \tag{7}\]  
The design of the 3D-SWZCC code requires three code sequences of 1D-SWZCC code, A, B, and C, where the first sequence encodes the spectral component; the second sequence encodes the temporal component; and the third sequence encodes the spatial component; their code lengths are \(L_{1}=K_{1}\), \(L_{2}=K_{2}\), and \(L_{3}=K_{3}\), respectively. \(K_{1}\), \(K_{2}\), and \(K_{3}\) express the supported cardinality in the spectral, time and space domains, respectively. Thus, the novel total code length and system capacity are \(L=L_{1}\cdot L_{2}\cdot L_{3}\) and \(K=K_{1}\cdot K_{2}\cdot K_{3}\), respectively. The 3D-SWZCC code sequence is shown below in Equation (8). Note that '\(g\)', '\(h\)', and '\(l\)' belong to \(\{1,2,\ldots,K_{1}\}\), \(\{1,2,\ldots,K_{2}\}\), and \(\{1,2,\ldots,K_{3}\}\), so that \(X_{g}\), \(Y_{h}\), and \(Z_{l}\) denote the \(g^{th}\), \(h^{th}\), and \(l^{th}\) code sequences of spectral component (\(X\)), temporal component (\(Y\)), and spatial component (\(Z\)), respectively. An example of 3D-SWZCC code for \(K_{1}=2\), \(K_{2}=3\), and \(K_{3}=3\) is presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \(Y_{0}=[1\) & \(0\) & \(0]\), \(Z_{0}=[1\) & \(0\) & \(0]\) \\ \hline \(X_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}\) & \(b_{1,1,1}=\begin{bmatrix}1&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(X_{1}=\begin{bmatrix}0\\ 1\end{bmatrix}\) & \(b_{2,1,1}=\begin{bmatrix}0&0&0&1&0&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(Y_{1}=\begin{bmatrix}0&0&1\end{bmatrix}\) & \(Z_{0}=[1&0&0]\) \\ \hline \(X_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}\) & \(b_{1,2,1}=\begin{bmatrix}0&0&1&1&0&0&0&0\\ 0&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(X_{1}=\begin{bmatrix}0\\ 1\end{bmatrix}\) & \(b_{2,2,1}=\begin{bmatrix}0&0&0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(Y_{2}=\begin{bmatrix}0&1&0\end{bmatrix}\) & \(Z_{0}=[1&0&0]\) \\ \hline \(X_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}\) & \(b_{1,3,1}=\begin{bmatrix}0&1&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(X_{1}=\begin{bmatrix}0\\ 1\end{bmatrix}\) & \(b_{2,3,1}=\begin{bmatrix}0&0&0&1&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(Y_{0}=\begin{bmatrix}1&0&0\end{bmatrix}\) & \(Z_{1}=[0&0&1]\) \\ \hline \(X_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}\) & \(b_{1,1,2}=\begin{bmatrix}0&0&0&1&0&0&0&0&1\\ 0&0&0&0&0&0&0&0\\ \end{bmatrix}\) \\ \hline \(X_{1}=\begin{bmatrix}0\\ 1\end{bmatrix}\) & \(b_{2,1,1}=\begin{bmatrix}0&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&1\end{bmatrix}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The 3D-SWZCC Code for \(K_{1}=2\), \(K_{2}=3\), and \(K_{3}=3\). There are eight distinct matrices \(A^{(d)}\), where \(d=\{1,2,3,4,5,6,7,8\}\). These are used to clearly explain the 3D-SWZCC code cross correlation [27]:  
\[A^{(d)}= \begin{cases}A^{(1)}=X^{T}YZ\\ A^{(2)}=X^{T}YZ\\ A^{(3)}=\overset{T}{X}YZ\\ A^{(4)}=\overset{-T}{X}\overset{-}{YZ}\end{cases}\begin{cases}A^{(5)}=X^{T}Y \overset{-}{Z}\\ A^{(6)}=X^{T}Y\overset{-}{Z}\\ A^{(7)}=\overset{-T}{X}\overset{-}{YZ}\end{cases}  \tag{8}\]  
\(\overset{-T}{X}\), \(\overset{-T}{Y}\), and \(\overset{-T}{Z}\) denote the complements of \(Y\), and \(Z\), respectively. The cross correlation between \(A^{(d)}\) and \(A_{g,h,l}\) can be expressed as [28]  
\[R^{(d)}(g,h,l)=\sum_{i=1}^{L_{1}}\sum_{j=1}^{L_{2}}\sum_{k=1}^{L_{3}}a^{(d)}_{i, j,k}a_{(i+a),(j+h),(k+l)} \tag{9}\]  
where \(a^{(d)}_{i,j}\) and \(a_{(i+g),(j+h),(k+l)}\) denote the elements of \(A^{(d)}\) and \(A_{g,h,l}\), respectively.