## 2 Processes of Constructing Extended SAC Labels  
In this section, we constructed extended SAC labels and analyzed their correlation properties. A SAC code family is characterized as (\(N\), \(\omega\), \(\lambda\)) where \(N\) is the code length, \(\omega\) is the code weight, and \(\lambda\) is the cross-correlation value. The extended SAC label \(\mathbf{E}(i,j)\) can be generated from MLS codes of {\(N\), (\(N+1\))/2, (\(N+1\))/4} according to the following mapping procedure:  
\[\mathbf{E}(i,j)\ =\ \mathbf{C}(i)\otimes\underbrace{\left[ \begin{matrix} \underline{0}\ldots\underline{0}&\underline{1}&\underline{0}\ldots\underline{ 0}\\ \end{matrix} \right]}_{\text{total }M\text{ matrix}},1<i<N,1<j<M \tag{1}\]  
where \(\mathbf{C}(i)\) is the \(i\)-th vector of MLS code, \(\underline{0}\) and \(\underline{1}\) are the vectors of all \(N\) elements being zeros and ones, \(\otimes\) is the Kronecker multiplication symbol, and \(M\) is the WDM channel number. Table 1 shows an exemplary label set of extended SAC labels with \(N=3\) and \(M=3\). The main idea is to divide the label vector into \(M\) blocks. For label \(\mathbf{E}(i,j)\), MLS code \(\mathbf{C}(i)\) is located in the \(j\)-th block, while other \(N\)(\(M\)-1) chips are all zeros. Compared with the conventional OCDMA labels, the code length of extended SAC labels is increased from \(N\) to \(L\), where \(L=M\)N.  
From the mapping procedure of (1), label length \(L\) can be varied by assigning different values to \(M\) and \(N\). We can adjust the available label number to match the connection number in GMPLS networks. For conventional SAC labels, the label numbers are limited to several discrete code lengths. For example, the code lengths of MLS codes are defined as \(2^{p}\)-1, where \(p\) is an integer larger than one. When MLS codes are used as optical labels, the label number for a network may be redundant or insufficient. On the other hand, the extended SAC labels are more flexible so that they can be applied to networks of different scales. Table 2 compares the available lengths of MLS and the proposed extended labels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Code Length**} & \multicolumn{3}{c}{**Number of Available Label Lengths**} \\ \cline{2-5}  & & **MLS Code** & **Extended SAC Label** \\ \hline \textless{}10 & 2 & 4 \\ \textless{}20 & 3 & 9 \\ \textless{}30 & 3 & 14 \\ \textless{}40 & 4 & 21 \\ \textless{}50 & 4 & 27 \\ \textless{}60 & 4 & 31 \\ \textless{}70 & 5 & 38 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of available label numbers of maximal length sequence (MLS) code and extended SAC label. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**\(j\)**} & \multicolumn{3}{c}{**\(i\)**} & \multicolumn{3}{c}{\(\mathbf{E}(i,j)\)**} \\ \hline \multirow{2}{*}{1} & 1 & 110 & 000 & 000 \\  & 2 & 101 & 000 & 000 \\  & 3 & 011 & 000 & 000 \\ \hline \multirow{2}{*}{2} & 1 & 000 & 110 & 000 \\  & 2 & 000 & 101 & 000 \\  & 3 & 000 & 011 & 000 \\ \hline \multirow{3}{*}{3} & 1 & 000 & 000 & 110 \\  & 2 & 000 & 000 & 101 \\ \cline{1-1}  & 3 & 000 & 000 & 011 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Extended spectral-amplitude-coding (SAC) label set of \(M=3\) and \(N=3\). Next, we investigated the correlation properties of \(\mathbf{E}(i,j)\) to evaluate the label quality effectively. Cross-correlation \(\lambda\) is defined as the number of overlapping chips between any two codes in a code family. As optical codes with large \(\lambda\) suffer serious PINN effects at the decoding end [15], \(\lambda\) should be minimized or reduced when OCDMA labels are designed. The correlation values between two extended SAC labels are defined as follows:  
\[\mathbf{E}(i,\ j)\odot\mathbf{E}(k,l)\ =\ \left\{ \begin{array}{ll}\frac{N+1}{2},&i \ =\ k,j\ =\ l.\\ \frac{N+1}{4},&i\
eq k,j\ =\ l.\\ 0,&\text{otherwise.}\end{array} \right. \tag{2}\]  
where \(\odot\) denotes the dot-product symbol [21].