Table 5 The monomial-walks, walk representations and path representations to the 18 design candidates

\(M_{1}^{Kw}\)

\(e_{2} e_{10}\)

\(W_{1} (M_{1}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{1}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{1}^{Kw} ) = \{ e_{2} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{1}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{1}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{2}^{Kw}\)

\(e_{4} e_{20}\)

\(W_{1} (M_{2}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{2}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{2}^{Kw} ) = \{ e_{4} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{2}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{2}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{3}^{Kw}\)

\(e_{1} e_{6} e_{10}\)

\(W_{1} (M_{3}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{3}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{3}^{Kw} ) = \{ e_{1} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{3}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{3}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{4}^{Kw}\)

\(e_{1} e_{8} e_{20}\)

\(W_{1} (M_{4}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{4}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{4}^{Kw} ) = \{ e_{1} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{4}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{4}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{5}^{Kw}\)

\(e_{3} e_{10} e_{16}\)

\(W_{1} (M_{5}^{Kw} ) =\left\{ {\begin{array}{*{20}l} {VS_{1} (M_{5}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{5}^{Kw} ) = \{ e_{3} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{5}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{5}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{6}^{Kw}\)

\(e_{3} e_{18} e_{20}\)

\(W_{1} (M_{6}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{6}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{6}^{Kw} ) = \{ e_{3} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{6}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{6}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{7}^{Kw}\)

\(e_{1} e_{5} e_{6} e_{10}\)

\(W_{1} (M_{7}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{7}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{7}^{Kw} ) = \{ e_{1} ,e_{5} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{7}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{7}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{8}^{Kw}\)

\(e_{2} e_{9} e_{10} e_{12}\)

\(W_{1} (M_{8}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{8}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{8}^{Kw} ) = \{ e_{2} ,e_{9} ,e_{12} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{8}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,SC_{2}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{8}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{9}^{Kw}\)

\(e_{1} e_{5} e_{8} e_{20}\)

\(W_{1} (M_{9}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{9}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{1}^{RR} ,{\kern 1pt} {\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{9}^{Kw} ) = \{ e_{1} ,e_{5} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{9}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{2}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{9}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{10}^{Kw}\)

\(e_{1} e_{7} e_{10} e_{16}\)

\(W_{1} (M_{10}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{10}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{10}^{Kw} ) = \{ e_{1} ,e_{7} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{10}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{10}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{11}^{Kw}\)

\(e_{3} e_{6} e_{10} e_{15}\)

\(W_{1} (M_{11}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{11}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{11}^{Kw} ) = \{ e_{3} ,e_{15} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{11}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{11}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{12}^{Kw}\)

\(e_{2} e_{9} e_{14} e_{20}\)

\(W_{1} (M_{12}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{12}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{12}^{Kw} ) = \{ e_{2} ,e_{9} ,e_{14} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{12}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{12}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{13}^{Kw}\)

\(e_{4} e_{10} e_{12} e_{19}\)

\(W_{1} (M_{13}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{13}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{12}^{Kw} ) = \{ e_{4} ,e_{19} ,e_{12} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{13}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{13}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{14}^{Kw}\)

\(e_{1} e_{7} e_{18} e_{20}\)

\(W_{1} (M_{14}^{Kw} ) =\left\{ {\begin{array}{*{20}l} {VS_{1} (M_{14}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,{\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{14}^{Kw} ) = \{ e_{1} ,e_{7} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{14}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{14}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{15}^{Kw}\)

\(e_{3} e_{8} e_{15} e_{20}\)

\(W_{1} (M_{15}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{15}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{15}^{Kw} ) = \{ e_{3} ,e_{15} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{15}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{15}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{16}^{Kw}\)

\(e_{3} e_{10} e_{16} e_{17}\)

\(W_{1} (M_{16}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{16}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{16}^{Kw} ) = \{ e_{3} ,e_{17} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{16}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{16}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{17}^{Kw}\)

\(e_{4} e_{14} e_{19} e_{20}\)

\(W_{1} (M_{17}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{17}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,{\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{17}^{Kw} ) = \{ e_{4} ,e_{19} ,e_{14} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{17}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{17}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)

\(M_{18}^{Kw}\)

\(e_{3} e_{17} e_{18} e_{20}\)

\(W_{1} (M_{18}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{18}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{18}^{Kw} ) = \{ e_{3} ,e_{17} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\)

\(VS^{P} (W_{1} (M_{18}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{2}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\)

\(A^{P} (W_{1} (M_{18}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\)