# Error-Correction Coding Using Polynomial Residue Number System
## 3 Theoretical Foundations of Building Redundant Codes of Polynomial Residue Number System
### PRNS Codes with One Control Base
Find the polynomial interval for the erroneous combination:  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]3  
It is known that orthogonal bases of redundant PRNS are defined as  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]4  
Substitute Expression (45) into Equality (44). We obtain  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]5  
The theorem is proved. \(\square\)  
**Example 3. **_Consider the application of Theorem 4 for the code of polynomial residue number system with informational bases \(p_{1}(z)=z+1\), \(p_{2}(z)=z^{2}+z+1\), \(p_{3}(z)=z^{4}+z^{3}+z^{2}+z+1\), \(p_{4}(z)=z^{4}+z^{3}+1\). The operating range of the code is \(W_{4}(z)=\prod\limits_{i=1}^{4}p_{i}(z)=z^{11}+z^{8}+z^{7}+z^{5}+z^{3}+z^{2}+z+1\). Here, we select \(p_{5}(z)=z^{4}+z+1\) as a control base._  
_Consider \(D(z)=z^{5}=(1,\ z+1,\ 1,\ z^{3}+z+1,\ z^{2}+z)\). The code combination for this polynomial is allowed, as \(\text{deg}D(z)<\text{deg}W_{4}(z)\)._  
_Let the error occur in the residue \(d_{1}(z)\) and let the error depth be \(\Delta d_{1}(z)=1\). Then, the erroneous combination is \(D^{*}(z)=(0,\ z+1,\ 1,\ z^{3}+z+1,\ z^{2}+z)\). Using Expression (41), the weight of the orthogonal basis is \(m_{1}(z)\) = 1; thus, we obtain_  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]6Calculating the polynomial interval using the CRT for polynomials and using the orthogonal bases from Example 2, we have  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]7  
In this case, \(S(z)=\left[\frac{D^{*}(z)}{W_{n}(z)}\right]=z^{3}+z^{2}+z\). It is possible to identify both a location of the erroneous residue of the code combination and an error depth using Theorems 1-3. It is necessary to find the corresponding polynomial interval in order to do this. Due to the minimal introduced redundancy, a collision may occur in the code of PRNS with one control base. In this case, errors on informational bases and errors on control bases may produce the same value of the polynomial interval. Let the error occur in the residue \(d_{5}(z)\) and let the error depth be \(\Delta d_{5}(z)=z^{2}+z+1\). Then, we have \(D^{*}(z)=(1,\,z+1,\,\,1,\,z^{3}+z+1,\,\,1)\). Using Expression (41), the weight of the orthogonal basis is \(m_{5}(z)=z\); thus, we obtain  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]8  
We have obtained a polynomial interval that coincides with the polynomial interval in the example. This is due to the fact that the number of possible polynomial intervals with one control base is equal to  
\[D_{iR}(z)-D_{iV}(z)=\left|R_{i}(z)-V_{i}(z)\right|_{p_{i}(z)}m_{i}(z)\frac{W_{n} (z)}{p_{i}(z)} \tag{26}\]9  
At the same time, the number of polynomial intervals that correspond to errors on the bases of the code of PRNS is determined by  
\[R_{i}(z)\
eq V_{i}(z)\text{mod}p_{i}(z)\]0  
It is obvious that \(N_{1}<N_{2}\). Therefore, codes of polynomial residue number system with one control base are only able to detect calculation errors. It is possible to eliminate this collision by increasing the introduced redundancy in PRNS.