## 4 Experimental Results
The 'Gain' in these tables denotes the ratio of message length of the proposed systems to the message length of the referenced system and is written as a percentage. The 'G1' and 'G2' are gains when the referenced systems are those in [4; 5], respectively. As can be seen in Tables 1 to 3, the minimum \(s\) of the proposed systems is always smaller than that of the referenced systems. The number of embedded messages in the proposed systems is also larger than that in the referenced systems. In Table 1, when the Lena image is considered, the proposed systems with an RS(15, 11) code and an RS(31, 23) show about three times more efficient transmission than the referenced system [4]. In Table 3, the proposed systems with an RS(15, 7) code and an RS(31, 15) are more than two times more efficient than the referenced systems [4; 5]. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Rate** & **Min. \(s\)** & **No. Messages** & **Gain (\%)** \\ \hline Ref. [PERSON] [4] & 1 & 16 & 1024 & G1 & G2 \\ Ref. [PERSON] [5] & 1 & 14 & 1296 & 293.3 & 231.8 \\ Pro. RS(15, 11) & 0.73 & 8 & 3004 & 293.3 & 260.2 \\ Pro. RS(31, 23) & 0.74 & 6 & 5360 & 523.5 & 413.6 \\ Pro. RS(31, 15) & 0.48 & 6 & 3496 & 341.4 & 269.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Minimum size of \(s\) for BER = 0 when the Lena image is used. G, gain. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Rate** & **Min. \(s\)** & **No. Messages** & **Gain (\%)** \\ \hline Ref. [PERSON] [4] & 1 & 12 & 1764 & G1 & G2 \\ Ref. [PERSON] [5] & 1 & 12 & 1764 & 170.3 & 170.3 \\ Pro. RS(15, 7) & 0.47 & 8 & 1911 & 108.4 & 108.4 \\ Pro. RS(31, 23) & 0.74 & 8 & 3039 & 172.3 & 172.3 \\ Pro. RS(31, 15) & 0.48 & 6 & 3496 To verify the performance of image recovery, PSNR performances for the three images in Figure 3 according to the embedding rate are shown in Figure 8. The 'Embedding rate' in Figure 8 represents the ratio of the number of embedded messages to the number of pixels in an embedded image. The 'Inf.' in Figure 8 denotes infinite PSNR. Infinite PSNR cannot be illustrated in these figures; however, for convenience, it is located at 103.4 dB, the MSE of which corresponds to \(\frac{1}{512\times 512}(\approx 3.815\times 10^{-6})\). Since the pixels in the images are represented with eight bits, MAX in (24) is 255. The 'Ref. [PERSON] (2015), Dec.' and 'Ref. [PERSON] (2015), Rec.' in Figure 8 denote PSNRs in directly decrypted image and recovered image of [PERSON]'s scheme in Zhang (2015), respectively. \(M\) and \(S\), which represent the number of LSBs of pixels and the number of hidden data per a group in Zhang (2015), are considered as two and one, respectively. The PSNRs in directly decrypted images and in recovered images represent those before image recovery and after image recovery in Zhang (2015), respectively, and PSNR performances in directly decrypted images were shown in Zhang (2015). In Figure 8, most embedding rates of the proposed systems are higher than those of the referenced systems, which satisfy the infinite PSNR. In Figure 8a, the proposed scheme with RS(31, 23) codes has an embedding range up to 0.02, which guarantees an infinite PSNR, while the referenced systems have an embedding range up to 0.007. Similarly, in Figure 8b, the proposed scheme with RS(31, 15) codes has an embedding range up to 0.014 for infinite PSNR.