## 4 Reed-Solomon Codes Effectiveness against Single-Frequency EMD
### Impact of the Symbol Size  
As evidenced in Figure 3, increasing the symbol size results in more CTP and CFN peaks. Equation (2) indicates that, in addition to CTP and CFN peaks at harmonics, there are always \(r-1\) CTP and CFN peaks between two harmonics of \(f_{\text{bit}}\). Take the symbol size of 4 as an example, i.e., Figure 3. In this case, the first groups of disturbance frequencies, at which a lot of CTP and CFN occur, are in the range of \(5/4\), \(6/4\), and \(7/4\) of the rate \(f_{\text{bit}}\) 200 MHz (i.e., 250 MHz, 300 MHz, and 350 MHz). The second groups are the disturbance frequencies in the range of the harmonics of the \(f_{\text{bit}}\) (i.e., 200 MHz and 400 MHz), when \(j\) is a multiple of \(m\) (i.e., symbol size) in Equation (2). The former results in all possible code words with one-symbol value except all 0's and all 1's code words (i.e., \(1\leq\text{symbol-value}\leq 2^{m}-1\)), e.g., \([0110,0110,0110,\ldots,0110,0110,0110]\); and the latter only yields all 0's and all 1's code words (i.e., symbol-value = 0 or \(2^{m}-1\)), e.g., \([1111,1111,1111,\ldots,1111,1111,1111]\). Note that, in both cases, all symbols of a code word turn into a single value. Moreover, this altered code word is considered valid based on the initially generated dictionary.  
Although smaller symbol sizes have less CTP and CFN peaks, they contain a wider frequency range (see, e.g., Figure 2). This is because of the smaller Hamming distance that makes the code words more susceptible to turn into another valid code word. In other words, smaller number of bit-flips is required to transform it into another valid code word. Thus, more frequencies around each peak can generate CTP or CFN. It is worth noting that all the corrupted code words generated by the frequencies around each peak and the peak itself are identical.  
In mathematical terms, the minimum Hamming distance among generated code words of the RS Codes dictionary is \(n-k+1\). Thus, at least \(n-k+2\) bit-flips are required to transform any code word to another valid code word.  
As shown in Figure 6, increasing the symbol size by keeping \(k\) constant (i.e., \(k=1\)) increases the minimum Hamming distance (i.e., \(n-k+1=n=2^{m}-1\)) as well as the imposed overhead (i.e., \(n=2^{m}-1\)). This also indicates an increase in the required number of bit-flips to generate a valid code word. However, due to the repetitive behavior of continuous EMD wave, the probabilities of generating one-symbol-value code words are higher. This is the reason why increasing the symbol size is also not in our favor and causes more peaks of CTP and CFN.  
It is worth noting that an increase in the symbol size also decreases the ratios of DTP (green category) and CFP (i.e., the category in a shade of orange) and also increases the ratio of CTN (i.e., the category in shade on orange). This indicates that, for larger symbol sizes, RS Codes have a stronger detection capability than correction capability. RS Codes are  
Figure 6: Symbol size effect on the minimum hamming distance while \(k=1\).  
designed to deal with bursts of errors. However, continuous wave EMDs occur in a periodic pattern, and this is the reason why data corruption is more intense for codes with greater block length. To this end, these setups lead to very low ratios of DTP (green category).