## 4 Reed-Solomon Codes Effectiveness against Single-Frequency EMD
### Peaks of the Undetected Corrupted Data  
Figures 2 and 3 depict the fault category distribution of RS Codes under the considered setup for the symbol sizes of 3 and 4, respectively. Figures 2a,b and 3a,b show this distribution for a message length of 1 symbol (i.e., \(k=1\)), while Figures 2c,d and 3c,d, under the same setup, represent the fault category distribution for a message length of three symbols (i.e., \(k=3\)). As can be seen in Figures 2 and 3, there are sudden increases in CFN and CTP at specific disturbance frequencies (\(f_{\text{EMD}}\)). These frequencies are dependent on the symbol size (\(r\)) and the bit-rate (\(f_{\text{bit}}\)), as shown in Equation (2). \[f_{\text{UCD}}=\frac{j}{r}\cdot f_{\text{bit}},\,j\in\mathbb{N}^{+}, \tag{2}\]  
Based on the generated dictionary, we know that there are two types of CFN/CTP that could be generated due to the EMD effect: CFN/CTP triggered by one-symbol-value code words (i.e., all symbols in a code word are identical) and CFN/CTP triggered by multi-symbol-value code words (i.e., all other code words except the one-symbol-value code words). Based on the specified setup parameters, code words with multi-symbol values are only generated when \(k>1\) (i.e., Figure 2c). In this case, in addition to one-symbol-value code words, the generated dictionary contains multi-symbol-value code words as well. Moreover, it is found that, owing to the periodicity of the single-frequency disturbances, all CFN and CTP at these peaks/frequencies are one-symbol-value code words. This is due to the fact that, at the mentioned EMD frequencies, sampling positions of all symbols are identical, which would alter them equally. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{3}{*}{G\({}_{1}\)} & **Symbol** & **Field** & **Block** & **Message** & **Correction** & **Hamming** \\  & **Size (\(m\))** & **Size (\(2^{m}\))** & **Length (\(n\))** & **Length (\(k\))** & **Capability (\(s\))** & **Distance (\(d\))** \\ \cline{2-6}  & 3 & 8 & 7 & 1 & 3 & 7 \\  & 4 & 16 & 15 & 1 & 7 & 15 \\ \cline{2-6}  & 3 & 8 & 7 & 3 & 2 & 5 \\  & 4 & 16 & 15 & 3 & 6 & 13 \\ \hline \end{tabular}
\end{table} 
Table 2: Reed–Solomon Codes setup parameters. Figure 1: Conceptual overview of the experimental setup. The areas in a single-frequency disturbance that lead to a bit-flip are demonstrated in Figure 4. As can be observed, the part of the sine wave (i.e., single-frequency disturbance) greater than or equal to 0.5 V is responsible for turning a bit to 1, while the negative part, where it is less than or equal to \(-0.5\) V, is responsible for turning a bit to 0. The middle area has no effect as it cannot create a bit-flip. Equation (3) calculates the domain (\(\Theta_{E}\)), which leads to half of all possible bit-flips, i.e., only the part where a bit turns to 1 or only the part where a bit turns to 0. \[ \begin{split}&\Theta_{E}=\Theta_{2}-\Theta_{1}\text{ and }\Theta_{1}=\pi-\Theta_{2}\rightarrow\Theta_{E}=\pi-\cdot\Theta_{1},\\ & A\cdot\sin(\Theta_{1})=0.5\rightarrow\Theta_{1}=arcsin(\frac {0.5}{A}),\\ &\Theta_{E}=\pi-2\cdot arcsin(\frac{0.5}{A})=2\cdot arccos(\frac {0.5}{A}),\end{split}  \tag{3}\]  
Using the information from Figure 4 and having the number of valid code words in the dictionary (i.e., \(q^{m^{k}}=2^{m^{k}}\)), the ratios of DTP, CTP, and CFN at the harmonics of the disturbance frequency (i.e., \(f_{\text{EMD}}=j\cdot f_{\text{bit}}\), \(j\in\mathbb{N}^{+}\)) can be calculated by Equations (4)-(6). Figure 2: Reed–Solomon Codes Effectiveness under single-frequency electromagnetic disturbances for the symbol size of 3. \[\text{DTP (\%)} =\frac{(2\pi-2\cdot\Theta_{E})}{2\pi}\cdot 100, \tag{4}\] \[\text{CTP (\%)} =2\cdot\frac{1}{2^{n}}\cdot\frac{\Theta_{E}}{2\pi},\] (5) \[\text{CFN (\%)} =([\frac{2^{m^{k}}-2}{2^{m^{k}}}\cdot\frac{2\cdot\Theta_{E}}{2\pi} ]+[2\cdot(\frac{1}{2^{m^{k}}}\cdot\frac{\Theta_{E}}{2\pi})])\cdot 100, \tag{6}\]  
As can be seen in Figure 4, the middle area of the graph has no effect, which, accordingly, results in a DTP.