Space-time block code designs based on algebraic field extension for full rate, large diversity product, and nonvanishing minimum determinant of codewords have received great attention. There are many different types of codes available for two-transmitter antennas, such as cyclotomic space-time block codes, the golden space-time block code, and rotation-based space-time block codes. In this paper, a more general space-time block code design scheme, which is called quadratic space-time block coding, is proposed for the two-transmitter antennas using quadratic field extension. The optimal design of the quadratic space-time block codes in terms of a diversity product criterion is also presented. It is shown that the optimal quadratic space-time block codes designed in this paper do not belong to the existing space-time block code family such as the cyclotomic, golden, and rotation-based space-time block codes. The simulation results demonstrate that the average codeword error rate of the optimal quadratic space-time block code attains about 0.5 dB signal to noise ratio gain over those of the optimal cyclotomic and golden space-time block codes.