# A New COVID-19 Detection Method Based on CSK/QAM Visible Light Communication and Machine Learning
## 4 Methodology
### Galois Field Mapping/Galois Fields Demapping  
This section describe how to convert a dsDNA image into a polynomial and vice versa as well as how to colour-code strands or dsDNA fragments using M-CSK/M-QAM modulation. A chromosome contains a single long molecule of DNA, only part of which corresponds to an individual gene. We developed a simple DNA-based model to represent the fields \(GF(p)\) and \(GF(p^{n}),n>1\). It is based on the differential migration of dsDNA fragments of different sizes in gel electrophoresis, which is a standard technique for dsDNA fragments of different sizes that have previously been obtained by PCR. Here, the size of a dsDNA fragment corresponds to the number of base pairs \([bp]\) that are contained in the fragment. Each element \(r\in GF(p)\) is represented by a dsDNA fragment whose size is unique to the element \(r\). Therefore, only \(p\) dsDNA fragments are necessary to represent all elements of \(GF(p)\). Table 1 shows this representation using dsDNA fragments of different sizes, where the smallest size \(S_{0}\) is composed of one or more genes and the largest is \(S_{p-1}\). Gel electrophoresis is used to visualize the DNA molecular representation of a nonzero element \(a^{k}\in GF(p^{n})\), which represents the coefficients of the polynomial expression given for Equation (20), as shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{r\in GF(p)}\) & **0** & **1** & **2** & **—** & \(\boldsymbol{p-1}\) \\ \hline Size of DNA Fragment [bp] & \(S_{0}\) & \(S_{1}\) & \(S_{2}\) & & \(S_{p-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DNA representation for elements in \(GF(p)\). The dsDNA fragments for each coefficient \(a_{i}\in GF(p),i=0,1,\ldots,n-1\) are loaded into different slots of the agarose gel matrix. The slots and their respective columns are numbered \(n-1,n-2,...,2,1,0\) according to the order of powers \(\alpha^{n-1},\ldots,\alpha^{2},\alpha,1\) from left to right. Then, an electric field is applied to force the molecules to migrate through the gel and be separated by size. \[\alpha^{k}=2\alpha^{n-1}+(p-1)\alpha^{n-2}+...+\alpha^{2}+(p-1)\alpha+(p-1). \tag{20}\]  
For this purpose, chains of size \(S_{2}\) were loaded into slot \(n-1\), chains of size \(S_{p-1}\) were loaded into slot \(n-2\), and from slot \(n-3\) to slot \(3\), chains of size \(S_{0}\) were loaded. Finally, chains of size \(S_{1}\) were loaded into slot \(2\), and chains of size \(S_{p-1}\) were loaded into slots \(1\) and \(0\). Thus, our model defines a unique DNA-based representation for each element of \(GF(p^{n})\). We should note that \(\alpha^{0}=1\), and the null element \(0\in GF(p^{n})\) does not have a representation as a power of \(\alpha\). Hence, the field \(GF(p^{n})\) has \(p^{n}\) elements, which are stored in a lookup table according to the power of each element. **Example 1**. : _To construct the field \(GF(2^{3}),n=3\) a new element \(\alpha\) is added to the field \(GF(2)\). \(\alpha\) is a root of the primitive polynomial \(P(x)=x^{3}+x+1\) with a degree of \(n=3\), which is used to generate the elements of \(GF(2^{3})\). Since \(\alpha\) is a root of the polynomial \(P(\alpha)=\alpha^{3}+\alpha+1=0\) then, \(\alpha^{3}=\alpha+1,\alpha^{4}=\alpha\alpha^{3}=\alpha(\alpha+1)\) and so on. The field \(GF(2^{3})\) has \(2^{3}=8\) elements._  
In Table 3, a \(\alpha=\alpha^{1}\) is introduced as a root, but \(\alpha^{0}=1\) and \(0\) can also be introduced, since they are do have representation in the field. In the case of a \(GF(2^{16})\) field containing \(2^{16}=256\) elements, the same method can be utilised to organise the elements for use in 16x16 MIMO arrays.