## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 2
\begin{array}{l}\mathbf{b}_{\text{BIP}}[c]+\mathbf{b}_{\text{BIP}}[c_{\text{ TC}}]=\left\{2\mathbf{b}_{\text{UNIP}}[c]-\mathbf{1}_{C}^{\forall}\right\}+ \left\{2\mathbf{b}_{\text{UNIP}}[c_{\text{TC}}]-\mathbf{1}_{C}^{\forall} \right\}\\ =2\cdot\left\{\mathbf{b}_{\text{UNIP}}[c]+\mathbf{b}_{\text{UNIP}}[c]\right\}- 2\cdot\mathbf{1}_{C}^{\forall}=2\cdot\mathbf{1}_{C}^{\forall}-2\cdot\mathbf{ 1}_{C}^{\forall}=0\end{array} \] (A36)  
where in the third equality we used the already established one's complementarity \(\mathbf{b}_{\text{UNIP}}[c]+\mathbf{b}_{\text{UNIP}}[c_{\text{TC}}]=\mathbf{1 }_{C}^{\forall}\) of UNIP eDACs. **(iii)**: The proof (185) \(\mathbf{b}[c_{\text{TC}(-1)}]=\
abla\mathbf{b}[c]\Leftrightarrow\
abla\mathbf{b }[C-c]=\
abla\mathbf{b}[c]\) stems from the result (ii) \(\mathbf{b}_{\text{UNIP}}[c]\ +\mathbf{b}_{\text{UNIP}}[c_{\text{TC}}]=\mathbf{1}_{S}^{\forall}\) (182), which in turn implies  
\[\mathbf{b}_{\text{UNIP}}[c]\ +\mathbf{b}_{\text{UNIP}}[c_{\text{TC}}]=\mathbf{1}_{B}^{ \forall},\ \ \mathbf{b}_{\text{UNIP}}[c+1]\ +\mathbf{b}_{\text{UNIP}}[(c+1)_{\text{TC}}]=\mathbf{1}_{B}^{\forall}\] (A37)The last line is equivalent to following lines:  
\[ \begin{array}{c}\therefore\mathbf{b}[c]\ +\mathbf{b}[c_{\intercal C}]=\mathbf{b}[c+1]\ + \mathbf{b}[(c+1)_{\intercal C}]\\ \therefore\mathbf{b}[c]\ +\mathbf{b}[c_{\intercal C}]=\mathbf{b}[c+1]\ +\mathbf{b}[c_{ \intercal C}-1]\\ \therefore\mathbf{b}[c_{\intercal C}]\mathbf{-b}[c_{\intercal C}-1]=\mathbf{b}[ c+1]-\mathbf{b}[c]\\ \therefore\mathbf{b}[c_{\intercal(C-1)}+1]-\mathbf{b}[c_{\intercal(C-1)}]= \mathbf{b}[c+1]-\mathbf{b}[c]\\ \therefore\mathbf{\
abla b}[c_{\intercal(C-1)}]=\
abla\mathbf{b}[c]\end{array}  \tag{108}\]  
where in the second line in (108) we used the fact that the ordering of indices reverses under reverse-operator reflections (proven using definition (167)):  
\[(c+1)_{\intercal C}=c_{\intercal C}-1. \tag{109}\]  
In the third line we rearranged the equation. In the fourth line the mirror-image index definitions were recalibrated from a \(C\)-vector to a \(C-1\)-vector (with \(C-1\) is the length of the step vectors, \(\
abla\mathbf{b}[c]\)), based on the following identities, stemming from definition (167) of mirror-image index:  
\[c_{\intercal C}=c_{\intercal(C-1)}+1,\ \ c_{\intercal C}-1=c_{\intercal(C-1)} \tag{110}\]  
The last line in (108) is our sought result (185) on centrosymmetry of the codeword steps. **(iv)**: write the mirror-image currents, using (2), and add up the two currents:  
\[ \begin{array}{c}J_{c}=\mathbf{b}^{T}[c]\mathbf{I},\ \ J_{c_{\intercal C}}= \mathbf{b}^{T}[c_{\intercal C}]\mathbf{I}\\ \therefore\ I_{c}+J_{c_{\intercal C}}=\mathbf{b}^{T}[c]\mathbf{I}+\mathbf{b}^{T}[ c_{\intercal C}]\mathbf{I}\\ =\left(\mathbf{b}^{T}[c]+\mathbf{b}^{T}[c_{\intercal C}]\right)^{T}\mathbf{I}= \left\{\begin{array}{c}(\mathbf{b}_{\text{UNIP}}[c]+\mathbf{b}_{\text{UNIP} }[c_{\intercal C}])\cdot\mathbf{I}=\mathbf{I}_{\text{S}}^{\forall}\cdot \mathbf{I}=I_{\text{tot}}\\ (\mathbf{b}_{\text{BIP}}[c]+\mathbf{b}_{\text{BIP}}[c_{\intercal C}])\cdot \mathbf{I}=\mathbf{0}\cdot\mathbf{I}=0\end{array}\right.\end{array}  \tag{111}\]  
To recapitulate,  
\[J_{c}+J_{c_{\intercal C}}=\left\{ \begin{array}{c}I_{\text{tot}},\ \text{for UNIP}\\ 0,\ \text{for BIP}\end{array} \right. \tag{112}\]  
Using Corollary V-(vii) the constancy of \(J_{c}+J_{c_{\intercal C}}\) implies that \(\mathbf{J}\) is a raised-skew-centrosymmetric vector with center \(\mathbf{J}=\frac{1}{2}I_{\text{tot}},\ \ \ \ \mathbf{J}\ =0\). **(v)**: Specialize \(\
abla J_{c}\equiv(\
abla\mathbf{b}[c])^{T}\mathbf{I}\) (145) to indices \(c\) and \(c_{\intercal(C-1)}=C-c\), yielding  
\[\
abla J_{c}=\
abla\mathbf{b}[c]\cdot\mathbf{I},\ \ \ \
abla J_{C-c}=\
abla \mathbf{b}[C-c]\cdot\mathbf{I}, \tag{113}\]  
which together with \(\
abla\mathbf{b}[c]\ =\
abla\mathbf{b}[C-c]\) (185) imply \(\
abla J_{c}=\
abla J_{C-c}=\
abla J_{\intercal(C-1)}\), proving the sought steps centrosymmetry (v). \(\square\)