We propose a novel precoded UWB (PUWB) system designed for multiple access (MA) communications. In our design, we try to enable concentration of the received signal power from different users (synchronized) at different time slots, thus make the matched filter (MF) based multiuser detection eligible for optimal reception. The optimum code is constructed to maximize the output signal-to-interference ratio (SIR). Specifically, we formulate the codeword design problem into an orthogonal basis finding problem, and the Gram-Schmidt algorithm can be easily applied. Computer simulations demonstrate the satisfactory performance of the proposed system. Extended simulations also cover the effect of symbol length; the resolution of the feedback code message; the number of multiple users; as well as the realizations from different standardized UWB channel models.