Recently, 2D versions of metamaterials, metasurfaces, have attracted more attention, [11] due to their advantages of low cost, low profile, and strong abilities to manipulate spatial and surface waves. Novel generalized sheet transition condition method [12] and transverse resonance method [13] were first presented to analyze the EM performance of metasurfaces. Then generalized Snell's law [14] was proposed to introduce the concept of abrupt phase when designing metasurfaces. By changing the size, shape, or orientation of unit cells, the abrupt phase provided by the metasurface can be tailored accordingly, and the outgoing EM waves are engineered arbitrarily. Metasurfaces have offered more convenience and freedom for manipulating EM wavefronts, and have been widely applied in the microwave, [15][16][17][18][19][20] terahertz, [21][22][23][24] visible, [25][26][27][28] and even acoustic [29,30] frequencies.Metamaterials and metasurfaces described by continuously effective medium parameters and phase distributions have powerful capabilities in controlling EM waves, but in static ways. That is to say, once a metamaterial or metasurface is fabricated, its function will be fixed. In order to reach real-time controls to EM waves, digital coding characterization has been proposed to describe metamaterial, resulting in the concepts of coding, digital, and programmable metamaterials. [31] The binary 1-bit digital codes "0" and "1" are adopted to indicate the reflection phases of 0° and 180°, from which one can manipulate EM waves using different coding sequences. The digital codes have been extended to 2-bit and more to bring more freedom for controlling scattering beams. The digital states "0." "1," "2," and "3" represent the reflection phases of 0°, 90°, 180°, and 270°, respectively.By designing a unit cell controllable by a diode to achieve either "0" or "1" state, the digital and programmable metamaterials have been realized to reach real-time manipulations to EM waves. [31] The digital coding representation links the traditional metamaterials to information theory, giving us an opportunity to control EM performance through discrete digital states. Based on these concepts, many kinds of functions such as beam steering [31][32][33] and reduction of radar crosssections [34] have been achieved by switching coding sequences on coding metamaterials in microwave and terahertz regions. Recently, the concept of anisotropic coding metamaterials has been demonstrated, which can achieve two independent coding behaviors for different polarizations. [35] Furthermore, convolution operations on coding metasurfaces were presented to Coding representation of metamaterials builds up a bridge between the physical world and the digital world, making it possible to manipulate electromagnetic (EM) waves by digital coding sequences and reach field-programmable metamaterials. Here, the coding space is extended to complex domain and proposed complex digital codes to provide closer essence of EM-wave propagation. Based on the analytic geometr...