# Transforming Group Codes in Mealy Finite State Machines with Composite State Codes
## 5 Example of Synthesis of Mealy FSM \(U_{3}\)
So, the PBFs representing _LUTer_ are multiplied by \(g_{k}(k\in\{1,\ldots,KC\})\). The resulting expressions represent SBFs (13) and (14). For our example, we show only the SOPs resulting from the partial functions \(y_{1}^{1}\) and \(D_{1}^{1},y_{1}^{2}\) and \(D_{1}^{2}\), and \(y_{1}^{3}\) and \(D_{1}^{3}\). These partial SOPs are represented by the following equations:  
\[ \begin{array}{l}D_{1}^{1}=F_{10}^{1}=g_{1}\tau_{3}\tau_{4}\tau_{6};\\ D_{1}^{2}=F_{4}^{2}\lor F_{8}^{2}=g_{2}\tau_{3}\bar{\tau}_{4}x_{3}\lor g_{2} \tau_{2}\tau_{4}\bar{x};\\ D_{1}^{3}=F_{3}^{3}\lor F_{8}^{3}=g_{3}\tau_{3}\tau_{4}\lor g_{3}\tau_{3} \tau_{4}x_{7}\lor g_{3}\tau_{4}x_{3}\bar{x}.\end{array}  \tag{23}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(a_{m}\) & \(CS(a_{m})\) & \(a_{S}\) & \(CC(a_{S})\) & \(X_{h}\) & \(D_{h}\) & \(Y_{h}\) & \(h\) \\ \hline \(a_{8}\) & 00 & \(a_{9}\) & 1001 & 1 & \(D_{1}D_{4}\) & \(y_{2}y_{4}\) & 1 \\ \hline \(a_{9}\) & 01 & \(a_{8}\) & 1000 & \(x_{7}\) & \(D_{1}\) & \(y_{3}\) & 2 \\  & & \(a_{7}\) & 0110 & \(\bar{x}_{7}\) & \(D_{2}D_{3}\) & \(y_{1}y_{2}\) & 3 \\ \hline \(a_{10}\) & 10 & \(a_{11}\) & 0111 & \(x_{3}\) & \(D_{2}D_{3}D_{4}\) & \(y_{2}y_{8}\) & 4 \\  & 10 & \(a_{5}\) & 0011 & \(\bar{x}_{3}x_{4}\) & \(D_{3}D_{4}\) & \(y_{5}y_{7}\) & 5 \\  & & \(a_{12}\) & 0011 & \(\bar{x}_{3}\bar{x}_{2}\) & \(D_{1}D_{3}D_{4}\) & \(y_{1}y_{5}\) & 6 \\ \hline \(a_{12}\) & 11 & \(a_{5}\) & 0011 & \(x_{4}\) & \(D_{3}D_{4}\) & \(y_{3}\) & 7 \\  & & \(a_{1}\) & 0000 & \(\bar{x}_{4}\) & – & – & 8 \\ \hline \end{tabular}
\end{table} 
Table 4: Table for _LUTer3_ of FSM \(A1\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(a_{m}\) & \(CS(a_{m})\) & \(a_{S}\) & \(CC(a_{S})\) & \(X_{h}\) & \(D_{h}\) & \(Y_{h}\) & \(h\) \\ \hline \multirow{4}{*}{\(a_{1}\)} & \multirow{4}{*}{00} & \(a_{4}\) & 0100 & \(x_{1}\) & \(D_{2}\) & \(y_{1}y_{2}\) & 1 \\  & & \(a_{3}\) & 0010 & \(\bar{x}_{1}x_{6}\) & \(D_{3}\) & \(y_{3}\) & 2 \\  & & \(a_{2}\) & 0001 & \(\bar{x}_{1}x_{6}\) & \(D_{4}\) & \(y_{2}y_{5}\) & 3 \\ \hline \multirow{4}{*}{\(a_{2}\)} & \multirow{4}{*}{01} & \(a_{2}\) & 0001 & \(x_{6}\) & \(D_{4}\) & \(y_{3}y_{4}\) & 4 \\  & & \(a_{5}\) & 0011 & \(\bar{x}_{6}\) & \(D_{3}D_{4}\) & \(y_{6}y_{7}\) & 5 \\ \hline \multirow{4}{*}{\(a_{3}\)} & \multirow{4}{*}{10} & \(a_{4}\) & 0100 & \(x_{1}\) & \(D_{2}\) & \(y_{7}\) & 6 \\  & & \(a_{2}\) & 0001 & \(\bar{x}_{1}x_{7}\) & \(D_{4}\) & \(y_{3}\) & 7 \\  & & \(a_{6}\) & 0101 & \(\bar{x}_{1}\bar{x}_{7}\) & \(D_{2}D_{4}\) & \(y_{1}y_{2}\) & 8 \\ \hline \multirow{4}{*}{\(a_{4}\)} & \multirow{4}{*}{11} & \(a_{6}\) & 0101 & \(x_{6}\) & \(D_{2}D_{4}\) & \(y_{3}\) & 9 \\  & & \(a_{9}\) & 1001 & \(\bar{x}_{6}\) & \(D_{1}D_{4}\) & \(y_{6}y_{7}\) & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: Table for _LUTer1_ of FSM \(A1\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(a_{m}\) & \(CS(a_{m})\) & \(a_{S}\) & \(CC(a_{S})\) & \(X_{h}\) & \(D_{h}\) & \(Y_{h}\) & \(h\) \\ \hline \multirow{4}{*}{\(a_{1}\)} & \multirow{4}{*}{00} & \(a_{4}\) & 0100 & \(x_{1}\) & \(D_{2}\) & \(y_{1}y_{2}\) & 1 \\  & & \(a_{3}\) & 0010 & \(\bar{x}_{1}x_{6}\) & \(D_{3}\) & \(y_{3}\) & 2 \\  & & \(a_{2}\) & 0001 & \(\bar{x}_{1}x_{6}\) & \(D_{4}\) & \(y_{2}y_{5}\) & 3 \\ \hline \multirow{4}{*}{\(a_{2}\)} & \multirow{4}{*}{01} & \(a_{2}\) & 0001 & \(x_{6}\) & \(D_{4}\) & \(y_{3}y_{4}\) & 4 \\  & & \(a_{5}\) & 0011 & \(\bar{x}_{6}\) & \(D_{3}D_{4}\) & \(y_{6}y_{7}\) & 5 \\ \hline \multirow{4}{*}{\(a_{3}\)} & \multirow{4}{*}{10} & \(a_{4}\) & 0100 & \(x_{1}\) & \(D_{2}\) & \(y_{7}\) & 6 \\  & & \(a_{2}\) & 0001 & \(\bar{x}_{1}x_{7}\) & \(D_{4}\) & \(y_{3}\) & 7 \\  & & \(a_{6}\) & 0101 & \(\bar{x}_{1}\bar{x}_{7}\) & \(D_{2}D_{4}\) & \(y_{1}y_{2}  
\[ \begin{array}{l}y_{1}^{1}=F_{1}^{1}\lor F_{8}^{1}=g_{1}\tau_{5}\tau_{4}x_{1}\lor g _{1}\tau_{5}\tau_{4}x_{1}\tau_{5};\\ y_{1}^{2}=F_{4}^{2}=g_{2}\tau_{3}\tau_{4}x_{3};\\ y_{1}^{3}=F_{3}^{3}\lor F_{6}^{3}=g_{3}\tau_{4}x_{4}\lor g_{3}\tau_{3}x_{4}x_{3} \tau_{4}.\end{array}