## 4 Theorems and Proofs
 \tag{85}\]  
The covariance matrix of (85) is as follows:  
\[\mathfrak{C}\left(\left(\frac{\
u_{j}}{c(\mathbf{x}_{j})+C(\Delta_{j})}\right) ^{T}C(\Delta_{j})\right)=\mathbb{E}\left(\left(\frac{\
u_{j}}{c(\mathbf{x}_{j })+C(\Delta_{j})}\right)^{T}C(\Delta_{j})\left(\left(\frac{\
u_{j}}{c( \mathbf{x}_{j})+C(\Delta_{j})}\right)^{T}C(\Delta_{j})\right)^{T}\right)=\left( \sigma_{\xi_{j}}^{2}\right)_{d} \tag{86}\]  
and  
\[\varsigma_{j,i}=\frac{U^{\prime}_{j,i}}{C(X_{j,i})+C(\Delta_{j,i})}C\big{(} \Delta_{j,i}\big{)}\in\mathbb{N}\left(0,\sigma_{\xi_{j,i}}^{2}=\mathfrak{C} \left(\frac{U^{\prime}_{j,i}}{C(X_{j,i})+C\big{(}\Delta_{j,i})}C\big{(}\Delta _{j,i}\big{)}\right)\right), \tag{87}\]  
along with  
\[\mathfrak{C}\left(\frac{\
u_{j,i}}{c(X_{j,i})+C(\Delta_{j,i})}C(\Delta_{j,i}) \right)=\mathbb{E}\left(\frac{\
u_{j,i}}{c(X_{j,i})+C(\Delta_{j,i})}C(\Delta_ {j,i})\left(\frac{\
u_{j,i}}{c(X_{j,i})+C(\Delta_{j,i})}C(\Delta_{j,i})\right) ^{T}\right)=\sigma_{\xi_{j,i}}^{2}. \tag{88}\]  
From (82) and (87) the quantities \(U_{j,i}\) and \(U^{\prime}_{j,i}\) are evaluated as follows:  
\[U_{j,i}=U^{\prime}_{j,i}-\frac{U^{\prime}_{j,i}}{C\big{(}X^{\prime}_{j,i})}C \big{(}\Delta_{j,i}\big{)}=U^{\prime}_{j,i}-\varsigma_{j,i}\in\mathbb{R}, \tag{89}\]  
and  
\[U^{\prime}_{j,i}=\frac{C(X^{\prime}_{j,i})}{C(X_{j,i})}U_{j,i}=U_{j,i}+\delta_ {j,i}\in\mathbb{R}. \tag{90}\]  
Let us denote by \(\
u\) the standard deviation of \(\overset{\rightarrow}{\delta}_{j}+\overset{\rightarrow}{\Upsilon}_{j}= \delta_{j,i}+\Upsilon_{j,i},0\leq i\leq d-1\), which is evaluated from (86) and \(\sigma_{\overset{\rightarrow}{\Upsilon}_{j}}^{2}\) as  
\[\
u=\sqrt{\left(\sigma_{\overset{\rightarrow}{\delta}_{j}}^{2}+\sigma_{ \overset{\rightarrow}{\Upsilon}_{j}}^{2}\right)_{d}}. \tag{91}\]  
The maximum-likelihood-based correction rules can be given in the form of:  
\[\mathbf{U}_{j}=\mathbf{A}:\frac{1}{\left(\pi 2\
u^{2}\right)^{d/2}}e^{-\frac{ \left\|\mathbf{U}^{\prime}_{j}-\mathbf{A}\right\|^{2}}{2\cdot 2}}\geq\frac{1}{\left(\pi 2\
u ^{2}\right)^{d/2}}e^{-\frac{\left\|\mathbf{U}^{\prime}_{j}-\mathbf{B}\right\|^{2 }}{2\cdot 2}}, \tag{92}\]  
and:  
\[\mathbf{U}_{j}=\mathbf{B}:\frac{1}{\left(\pi 2\
u^{2}\right)^{d/2}}e^{-\frac{ \left\|\mathbf{U}^{\prime}_{j}-\mathbf{A}\right\|^{2}}{2\cdot 2}}\leq\frac{1}{\left(\pi 2\
u^{2}\right)^{d/2}}e^{-\frac{ \left\|\mathbf{U}^{\prime}_{j}-\mathbf{B}\right\|^{2}}{2\cdot 2}}. \tag{93}\]  
The error probability for the case of decoding vector \(\mathbf{U}_{j}=\mathbf{A}\), is  
\[\Pr_{\mathfrak{C}}\left(\left\|\overset{\rightarrow}{\delta}_{j}+\overset{ \rightarrow}{\Upsilon}_{j}\right\|^{2}>\left\|\left(\mathbf{A}+\overset{ \rightarrow}{\delta}_{j}+\overset{\rightarrow}{\Upsilon}_{j}\right)-\mathbf{B} \right\|^{2}\right)=\Pr_{\mathfrak{C}}\left(\left(\mathbf{A}-\mathbf{B}\right)^ {T}\left(\overset{\rightarrow}{\delta}_{j}+\overset{\rightarrow}{\Upsilon}_{j} \right)<-\frac{\left\|\mathbf{A}-\mathbf{B}\right\|^{2}}{2}\right). \tag{94}\]For the case of correction of \(\mathbf{U}_{j}=\mathbf{B}\), the error probabilities are evaluated as  
\[\Pr_{e}\Bigg{(}\|\overset{\rightarrow}{\delta}_{j}+\overset{\rightarrow}{\Upsilon }_{j}\|^{2}>\|\bigg{(}\mathbf{B}+\overset{\rightarrow}{\delta}_{j}+\overset{ \rightarrow}{\Upsilon}_{j}\bigg{)}-\mathbf{A}\|^{2}\Bigg{)}=\Pr_{e}\Bigg{(}( \mathbf{B}-\mathbf{A})^{T}\bigg{(}\overset{\rightarrow}{\delta}_{j}+\overset{ \rightarrow}{\Upsilon}_{j}\bigg{)}<-\frac{\|\mathbf{B}-\mathbf{A}\|^{2}}{2} \Bigg{)}. \tag{95}\]  
The decision regions can be separated into two hyperplanes \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) along \(\mathbf{B}-\mathbf{A}\), which separate \(\mathbf{U}_{j}=\mathbf{A}\) and \(\mathbf{U}_{j}=\mathbf{B}\). In other words, the correction-condition of a given noisy \(\mathbf{U}^{\prime}_{j}\) is reduced to the following decision problem:  
\[\mathbf{U}_{j}=\left\{ \begin{array}{l}\mathbf{A},\,\text{if}\,\,\mathbf{U} ^{\prime}_{j}\in\mathcal{H}_{1},\\ \mathbf{B},\,\text{if}\,\,\mathbf{U}^{\prime}_{j}\in\mathcal{H}_{2}.\end{array} \right.