## 2 Design of Matrix-Type Coding Metasurface
### Matrix-Type Random Coding Theory and RCS Reduction Analysis
The general RCS expression of the scattering surface can be expressed by [46]:  
\[\sigma=\lim_{r\rightarrow\infty}4\pi r^{2}\frac{S_{s}}{S_{i}}=\lim_{r \rightarrow\infty}4\pi r^{2}\left|\frac{E_{s}}{E_{i}}\right|^{2}=\lim_{r \rightarrow\infty}4\pi r^{2}\left|\frac{H_{s}}{H_{i}}\right|^{2} \tag{3}\]  
where \(S_{i}\) and \(S_{s}\) are the energy density of incidence and scattering, respectively; \(|E_{i}|\) and \(|E_{s}|\) are the amplitudes of incident and scattered electric fields, respectively; and \(|H_{i}|\) and \(|H_{s}|\) are the amplitudes  
Figure 1: Matrix-type random coding flowchart. of the incident and scattered magnetic fields, respectively. The general RCS can also be expressed in the form of \(dBsm\):  
\[\sigma(dBsm)=10\lg\left|\sigma(m^{2})\right|=10\lg\left[\underset{r\to \infty}{\lim}4\pi r^{2}\left|\frac{E_{s}}{E_{i}}\right|^{2}\right] \tag{4}\]  
In practical military applications, the incident wave can be equivalent to the plane wave because of the transmitting and receiving sources are far from the target. Thus, \(\sigma\) and \(r\) are independent of one another. The RCS reduction compared to perfect electric conductor is represented by:  
\[\sigma(dBsm)_{r}=10\lg\left[\frac{\underset{r\to\infty}{\lim}4\pi r^{2}\left| \frac{E_{s}}{E_{i}}\right|^{2}}{\underset{r\to\infty}{\lim}4\pi r^{2}\left(1 \right)^{2}}\right]=10\lg\left[\left|\frac{E_{s}}{E_{i}}\right|^{2}\right] \tag{5}\]  
For a matrix-type random coding MS, each kind of basic unit occupy the half area of total surface in the case of \"0\" and \"1\" with same number. The total reflection coefficient can be approximated as the average reflection coefficient of both elements. The RCS reduction can be approximately expressed as [46]:  
\[\sigma(dBsm)_{r(1)}=10\lg\left[\frac{A_{0}e^{i\theta_{0}}+A_{1}e^{i\theta_{1} }}{2}\right]^{2} \tag{6}\]  
where \(A_{0}\) and \(A_{1}\) are the reflection coefficient amplitudes of basic units \"0\" and \"1\", respectively. \(\theta_{0}\) and \(\theta_{1}\) are the reflection phases of two basic units. The ratio of \"0\" and \"1\" basic units is introduced, which is defined as \(\alpha=m_{0}/m_{1}\), where \(m_{0}\) and \(m_{1}\) are the number of \"0\" and \"1\" units, respectively. Further, we introduce \(\alpha\) into the formula to express the RCS reduction characteristic with different ratio. However, it can only be used as a qualitative comparison, not a quantitative representation of RCS value. The RCS reduction of 1-bit coding MS under normal incidence can be approximated as:  
\[\sigma(dBsm)_{r(\alpha)}=10\lg\left[\frac{\alpha A_{0}e^{i\theta_{0}}+A_{1}e^ {i\theta_{1}}}{2}\right]^{2} \tag{7}\]  
As shown in Equation (7), once basic units of \"0\" and \"1\" are selected, the reflection coefficient magnitude and phase can be determined. Therefore, if the ratio \(\alpha\) is defined as a constant, the RCS reduction of MS will be fixed. In other words, for fixed \(\alpha\), the magnitudes of RCS reduction with different coding sequences are basically consistent under normal incidence. The ratio, scattered magnitude and phase play important roles in RCS reduction. This assumption is verified by furthering simulation and experiment.