## 3 3D-VWZCC Code Construction
### 3D-VWZCC Code Construction  
Let X, Y, and Z be three spectral, time, spatial code sequences, with \(M\), \(N\), \(P\), which are the code lengths associated by each sequence, respectively, and given as \(M=K_{1}\times W_{1}\), \(N=K_{2}\times W_{2}\), \(P=K_{3}\times W_{3}\), where the combination \((K_{i},W_{i})\) is a number of users and code \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(VWZCC=\left[\begin{array}{cccccc}\mathbf{1}&\mathbf{1}&0&0&0&0&0&0&0&0&0 \\ 0&0&\mathbf{1}&\mathbf{1}&0&0&0&0&0&0&0\\ 0&0&0&0&\mathbf{1}&\mathbf{1}&\mathbf{1}&0&0&0&0\\ 0&0&0&0&\mathbf{1}&\mathbf{1}&0&0&0\\ 0&0&0&0&0&0&0&\mathbf{1}&\mathbf{1}&0&0\\ 0&0&0&0&0&0&0&0&\mathbf{1}&\mathbf{1}&\mathbf{1}\\ \hline\hline\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 1D-VWZCC when \(K\) = 5 and \(W\) = 2.  
weight of each component. An example of the 3D-VWZCC code is shown in Table 3, when \(K_{1}=2\), \(K_{2}=K_{3}=3\)  
The relation of building the three-dimensional code is expressed by the following Equation [10; 11; 12; 13]:  
\[A_{g,h,l}=X_{g}^{T}Y_{h}Z_{l} \tag{12}\]  
where \(X_{g}^{T}\) is the \(g^{th}\) sequence code of \(X\), \(Y_{h}\) is the \(h^{th}\) sequence code of \(Y\), and \(Z_{l}\) is \(l^{th}\) sequence code of \(Z\), where \(g=1\), \(2\), \(3\),..., \(K_{1}\), \(h=1\), \(2\), \(3\),..., \(K_{2}\) and \(l=1\), \(2\), \(3\),..., \(K_{3}\). The total capacity of the 3D-VWZCC code is expressed by the relation below; it depends on the number of users of \(X\), \(Y\), and \(Z\)[10; 11; 12]:  
\[K=K_{1}\times K_{2}\times K_{3} \tag{13}\]  
To clarify the 3D-VWZCC code property, the characteristic matrices \(A^{(d)}=A^{(0)}\), \(A^{(1)},\ldots A^{(7)}\) are given as [13]:  
\[A^{(d)}=\left\{ \begin{array}{c}A^{(0)}=X^{T}YZ\\ A^{(1)}=X^{T}\overline{Y}Z\\ A^{(2)}=\overline{X}^{T}YZ\\ A^{(3)}=\overline{X}^{T}\overline{Y}Z\end{array} \right.\left\{ \begin{array}{ c}A^{(4)}=X^{T}\overline{Y}\overline{Z}\\ A^{(5)}=X^{T}\overline{Y}\overline{Z}\\ A^{(6)}=\overline{X}^{T}\overline{Y}\overline{Z}\\ A^{(7)}=\overline{X}^{T}\overline{Y}\overline{Z}\end{array} \right. \tag{14}\]  
\(X^{T}\) is the transpose of the matrix \(X\). \(\overline{X}\), \(\overline{Y}\), and \(\overline{Z}\) are the complements of the sequences codes of the matrices \(X\), \(Y\), and \(Z\), respectively. An example of the 3D-VWZCC code is represented in Figure 1, and it presents a three-dimensional graphical presentation according to the first code of VWZCC indicated in Table 3 where the weight of the spectral, temporal, and spatial sequences are, respectively, equal to \(W_{1}=W_{2}=W_{3}=2\). Figure 1: The 3D-VWZCC (spectral/time/spatial) code for the first users. Consequently, the 3D-VWZCC cross-correlation property between \(A^{(d)}\) and \(A_{g,h,l}\) is expressed by the following equation, where \(a^{(d)}{}_{i,j,l}\) is the component of \(A^{(d)}\) and \(b_{i,j,l}\) is the components of \(A_{g,h,l}\)[10; 11; 12]. \[R^{(d)}(g,h,l)=\sum_{i=0}^{M-1}\sum_{j=0}^{N-1}\sum_{l=0}^{P-1}a^{(d)}{}_{i,j,l }b_{i,j,l} \tag{16}\]  
Further, Table 4 shows the results of the cross-correlation of the 3D-VWZCC code and contains eight cases according to the characteristic matrices \(A^{(d)}\). According to Table 3, all cases give the same cross-correlation value, taking into account the intersection of g, h and l. We notice that the values of cases 0 to 3 are the same values of cases 4 to 7, and this is due to the orthogonality of the VWZCC codes. Moreover, the value \(W_{1}W_{2}W_{3}\) is varied and depends on the weight \(W_{1}\), \(W_{2}\) and \(W_{3}\) of each spectral, temporal, and spatial component, respectively; as a result, the 3D-VWZCC code has good flexibility. By compensating the characteristic matrices \(A^{(d)}\) of Equation (1) in Equation (2), we obtain Table 4, which gives the values of \(R^{(d)}\) that are equal to zero, except that \(R^{(0)}\) and \(R^{(4)}\) are written by the following equations:  
\[R^{(0)}(g,h,l)=\left\{ \begin{array}{cc}1&if\;\;g=0\cap h=0\cap l=0\\ 0&else\end{array} \right.