## 4 Modified Modular Projection Method with MLD  
Now consider a new method combining the advantages of the CRT, CRTf, and maximum likelihood projections [19]. The proposed modification concerns error localization and restoration of a correct number in the WNS; error detection is identical for the original method [19] and the modified method proposed below. In this regard, the description of error detection is omitted. According to [19], each projection is restored in the WNS using the CRT. The resulting projections are employed to calculate the values of the missing remainders. Next, the Hamming distances are calculated for each projection, and the correct projection is selected (Figure 1a). Reconstructing the weighted representation of projections using the CRT has high time delays. First of all, the delay is due to executing operations on a large modulus comparable to the dynamic range of the RRNS. Figure 1: _Cont._  
In the original method [19], the reconstructed weighted representations of modular projections are used, on the one hand, to calculate the missing remainders subsequently and, on the other hand, to obtain the correct number in the WNS. (One of the projections will be the correct number represented in the WNS.) In the modified method, we propose using (a) the ranks of the modular projections to obtain the missing remainders and (b) the approximate weighted characteristics of the CRTf to restore the modular projections in the WNS. This approach allows for calculating the missing remainders and reconstructing projections in the WNS in parallel (Figure 1b), not sequentially, as was proposed in [19]. The efficiency of the proposed modification largely depends on the efficiency of calculating the rank of a number represented in the RRNS. The choice of the CRTf for reconstructing modular projections in the WNS is not accidental: the approximate weighted characteristic \(F(X)\) closely relates to the rank \(r_{X}\) of the number \(X\) written in the RRNS. According to (3), the rank \(r_{X}\) is given by  
\[r_{X}=\frac{\sum_{i=1}^{k}\frac{M_{K}}{m_{i}}\Big{|}\frac{m_{i}}{M_{K}}\Big{|} _{m_{i}}x_{i}-X}{M_{K}}=\frac{\sum_{i=1}^{k}\frac{M_{K}}{m_{i}}\Big{|}\frac{m_ {i}}{M_{K}}\Big{|}_{m_{i}}x_{i}}{M_{K}}-\frac{X}{M_{K}}=\sum_{i=1}^{k}\frac{| m_{i}/M_{K}|_{m_{i}}x_{i}}{m_{i}}-\frac{X}{M_{K}}.\]  
Due to (4) and (5),  
\[r_{X}=\sum_{i=1}^{k}\frac{|m_{i}/M_{K}|_{m_{i}}x_{i}}{m_{i}}-\left|\sum_{i=1} ^{k}\frac{|m_{i}/M_{K}|_{m_{i}}x_{i}}{m_{i}}\right|_{1}.\]  
Recall that \(|\bullet|_{1}\) denotes taking the fractional part of a number. Hence, the rank of \(X\) is given by  
\[r_{X}=\lfloor\sum_{i=1}^{k}\frac{|m_{i}/M_{K}|_{m_{i}}x_{i}}{m_{i}}\rfloor= \lfloor\sum_{i=1}^{k}k_{i}^{*}x_{i}\rfloor,k_{i}^{*}=\frac{|m_{i}/M_{K}|_{m_{ i}}}{m_{i}}. \tag{10}\]  
Figure 1: Modular projection methods with MLD: (**a**) original, (**b**) modified. Note that \(F(X)\) and \(r_{X}\) are the integer and fractional parts of the same value. We introduce the notation  
\[F^{\prime}(X)=\sum_{i=1}^{k}k_{i}^{*}x_{i},k_{i}^{*}=\frac{|m_{i}/M_{K}|_{m_{i}}} {m_{i}}. \tag{11}\]  
Calculating \(F(X)\), we simultaneously find the weighted characteristic \(F(X)\) (Equation (5)) and the rank \(r_{X}\) of \(X\) (Equation (10)). Similar to the case of the CRTf (Equations (6) and (7)), we propose using a binary shift to implement the calculation of \(F^{\prime}(X)\) on a hardware base not supporting fractions. The accuracy of the constants \(k_{i}^{*}\) in (11) is estimated by analogy with [20]. Then  
\[F^{\prime}(X)=\sum_{i=1}^{k}k_{i}^{*}x_{i},k_{i}^{*}=\frac{|m_{i}/M_{K}|_{m_{i} }}{m_{i}}\cdot 2^{N}, \tag{12}\]  
where \(N=\log_{2}\left(M_{K}\sum_{i=1}^{k}(m_{i}-1)\right)\)[20]. The value \(F^{\prime}(X)\) is especially convenient for the hardware implementation of the error correction procedure for modular codes.