Summary: In this paper, we present the methods of construction of quaternary orthonormal system (so-called QU-system) with piecewise polynomials, discuss the feasibility of the construction methods, and obtain a set of explicit expressions of QU-system with degrees 1 to 3. And then, we investigate the properties of QU-system and the relationship between binary U-system and QU-system, and present the formulae of its basis value and Fourier-QU coefficient. Applying the investigated construction methods, we can construct a class of complete orthonormal system in \(\mathbf{L}^2[0,1]\), which contains both continuous and discontinuous piecewise polynomials. So QU-system has the properties of both Fourier trigonometric functions and Walsh functions. Finally, we apply numerical experiments to confirm that the convergence rate of Fourier-QU series is better than that of Fourier series, Walsh series and Fourier-BU series, if using the first finite terms of them to approximate the functions of one variable.