## 4 Experimental Results
For some value of \(s\), the BER of [PERSON]'s system is better than the BER of the proposed system without RS codes, since the embedding process is different, and the fluctuation functions in [PERSON]'s system are based on the difference of neighboring pixels. Figure 5: Example of error recovery by using RS codes in the Lena image. (**a**) Error pattern in Figure 4c; (**b**) Analysis of the codeword for the error pattern. Figure 6: _Cont._  
To investigate the performances of the proposed systems with RS codes, four RS codes are used, which are an RS(15, 11), an RS(15, 7), an RS(31, 23) and an RS(31, 15) code in this simulation. The code rates of the four RS codes are 0.73, 0.47, 0.74 and 0.48, respectively, and they correspond to around 1/2 and 3/4. The lengths of the codes are 15 and 31. These RS codes can be used for estimating the performance change according to the length and rate of RS codes. In Figure 7, BER performances of the referenced systems and the proposed systems are shown according to the size of \(s\) when the Lena, Peppers, and Jet images in Figure 3 are considered. The 'Ref. [PERSON] [4]' and 'Ref. [PERSON] [5]' in these figures also denote BERs for [PERSON]'s scheme in [4] and [PERSON]'s scheme in [5], respectively. The notations 'Pro. RS(15, 11),' Pro. RS(15, 7),' Pro. RS(31, 23),' and 'Pro. RS(31, 15)' represent the proposed systems with RS(15, 11), RS(15, 7), RS(31, 23) and RS(31, 15) codes, respectively. For most values of \(s\), the BER performances of the proposed systems with four RS codes are better than those of the referenced systems [4; 5]. In Figure 7c, it is shown that the BER performance of RS(15, 11) fluctuates as \(s\) increases, since the RS codes can correct up to two error symbols. For \(s=14\), 18 and 20, there are less than three error symbols in every codeword of RS(15, 11) codes, and the BER is zero. In Figure 7a,b, the BER of the proposed system is 0 for \(s=8\), though the BER of the referenced systems [4; 5] is non-zero. In Figure 7c, BERs of the proposed systems with RS(15,7) and RS(31,15) codes are better than the two referenced systems. If \(s\) is larger than or equal to 10, BERs of the two proposed systems are zero over the weakly correlated image. For high rate  
Figure 6: BER performances of referenced and proposed fluctuation functions for the three images. (a) Lena; (b) Peppers; (c) Jet. RS codes, such as RS(15,11) and RS(31, 23), BER performances in Figure 7c turn around as \(s\) increases, since the image is weakly correlated, and the error capability of the codes is small. Since the proposed systems consider RS codes with code rates of 0.75 and 0.5, the transmission efficiencies of the proposed systems are less than those of the referenced systems. Therefore, for fixed  
Figure 7: BER performances of the proposed reversible data-hiding (RDH) systems with Reed–Solomon (RS) codes for the three images. (**a**) Lena; (**b**) Peppers; (**c**) Jet. BER, the effective length of the embedded message must be considered for proper transmission. For fair comparison, the number of messages for BER = 0 is considered, where BER = 0 means that embedded messages are extracted without error. The main results of the proposed systems and the referenced systems [4; 5] are listed in Tables 1-3 for BER = 0 when the Lena, Peppers, and Jet images are considered, respectively. The 'Ref. Zhang [4],' 'Ref. Hong [5],' and 'Pro. RS(15, 11), Pro. RS(15, 7), Pro. RS(31, 23), Pro. RS(31, 15),' in Tables 1 to 3 correspond to the referenced systems [4; 5] and the proposed system with RS(15, 11), RS(15, 7), RS(31, 23) and RS(31, 15) codes, respectively. The 'Rate' in these tables denotes the code rates of the considered systems; since the referenced systems have no RS codes, their code rates are one. The 'Min. \(s\)' in these tables stands for the minimum size of \(s\) that guarantees BER = 0. The 'No. messages' in these tables denotes the number of actual embedded messages corresponding to minimum \(s\).