## 5 EMD-Resilient Reed-Solomon Codes
### Overall Performance of the Proposed Layer
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Setup Parameters**} & **DTP** & **DTN** & **DFP** & **CTP** & **CTN** & **CFP** & **CFN** \\ \hline \multirow{8}{*}{**Fact**} & \(k=1\mid r=3\mid A=1\) V & 0.99 & 0 & 7.38 & 1.38 & 82.2 & 4.98 & 3.06 \\  & \(k=1\mid r=3\mid A=3\) V & 0.21 & 0 & 0.59 & 0.82 & 90.54 & 4.08 & 3.76 \\  & \(k=3\mid r=3\mid A=1\) V & 0.97 & 0 & 0.98 & 0.03 & 94.51 & 1.68 & 1.84 \\  & \(k=3\mid r=3\mid A=3\) V & 0.21 & 0 & 0.04 & 0.01 & 95.66 & 0.18 & 3.91 \\  & \(k=1\mid r=4\mid A=1\) V & 0.65 & 0 & 2.46 & 0.43 & 92.97 & 1.28 & 2.18 \\  & \(k=1\mid r=4\mid A=3\) V & 0.21 & 0 & 0.10 & 0.28 & 95.28 & 0.98 & 3.14 \\  & \(k=3\mid r=4\mid A=1\) V & 0.65 & 0 & 0.31 & 0.0 & 97.20 & 0.1 & 1.82 \\  & \(k=3\mid r=4\mid A=3\) V & 0.21 & 0.0 & 0.0 & 0.0 & 97.20 & 0.0 & 2.59 \\ \hline \multirow{8}{*}{**Fact**} & \(k=1\mid r=3\mid A=1\) V & 0.99 & 0.03 & 7.35 & **0.08** & **83.40** & 8.15 & **0** \\  & \(k=1\mid r=3\mid A=3\) V & 0.21 & 0.00 & 0.29 & **0.01** & **94.12** & 5.37 & **0** \\ \cline{1-1}  & \(k=3\mid r=3\mid A=1\) V & 0.97 & 0 & 0.98 & **0.01** & **96.26** & 1.76 & **0.02** \\ \cline{1-1}  & \(k=3\mid r=3\mid A=3\) V & 0.21 & 0 & 0.04 & **0.00** & **99.45** & 0.23 & **0.06** \\ \cline{1-1}  & \(k=1\mid r=4\mid A=1\) V & 0.65 & 0.01 & 2.70 & **0.05** & **95.05** & 1.54 & **0** \\ \cline{1-1}  & \(k=1\mid r=4\mid A=3\) V & 0.21 & 0 & 0.16 & **0.00** & **98.66** & 0.97 & **0** \\ \cline{1-1}  & \(k=3\mid r=4\mid A=1\) V & 0.65 & 0.0 & 0.31 & **0.0** & **99.01** & 0.02 & **0** \\ \cline{1-1}  & \(k=3\mid r=4\mid A=3\) V & 0.21 & 0 & 0 & **0** & **99.79** & 0.0 & **0** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison between the original RS Codes and the proposed version (in %). Nevertheless, as can be seen in Figure 9 and 10, there are still several incidents of one-symbol-value CTP around some frequencies of \(f_{\text{UCD}}\) (e.g., around 200 MHz and 400 MHz in Figure 9d, as well as around 200 MHz, 300 MHz, and 400 MHz in Figure 10a). These scenarios could happen when the channel is in control but in a way that it does not control all symbols of a code word. Accordingly, the inverted symbol could remain untouched during the transmission. As a result, the data consumer receives a code word that is not within the initial generated dictionary; therefore, it can pass through the inversion layer without detection. Thereby, after reverting the applied changes, the received code word could turn into a valid code word for RS Codes, which results in a CTP or CFN. In general, it is possible to determine which code words can end up as a one-symbol-value CTP or CFN. Based on RS Codes parameters, the total number of possible corrupted code words at the consumer side is \((q^{m})^{n}\). Given that, the number of possible code words that can turn into one-symbol-value CFN while reverting changes is \(q^{m}-1\), if the transmitted code word is a one-symbol-value code word and is \(q^{m}\) (i.e., the total number of one-symbol-value code words), and if the transmitted code word being a multi-symbol-value code word. Similarly, the number of possible code words that can turn into one-symbol-value CTP while reverting changes is 1, which only happens when the transmitted code word is the same one-symbol-value code word as the input. For instance, Table 4 shows the possible cases that can turn into a one-symbol-value CFN or CTP when \(r=3\), \(k=3\), and the transmitted code word is \([2,2,2,2,2,2,2]\). As can be seen, seven possible cases (i.e., \(q^{m}-1\)) lead to CFN, and one case leads to CTP. Note that this behavior holds for all other one-symbol-value code words regardless of their initiated setup parameters. Figure 10: The impact of the inversion layer on the EM-resiliency of RSCodes for a symbol size of 4.