In some visible light communication (VLC) systems, transmitters are light sources which support not only illumination but also information broadcast and positioning. Since transmit signals in these systems inevitably interfere with each other at the receiver side, extra devices may be added to synchronize the transmitters, but this impairs an appealing advantage in VLC systems that existing lighting infrastructure can be simply reused. This paper proposes a novel multiple access scheme to realize decentralized asynchronous VLC systems such that both the transmitters and receivers are asynchronous. For a system with N transmitters, two unique codewords of L chips are allocated to each transmitter where L is a prime number larger than 4N. Those codewords are designed to help maintain a constant transmitted average power and enhance channel estimation, as well as conveying data. Our scheme enables the receiver to obtain one channel estimate and one data symbol per L chips. Simulation results show that the proposed scheme significantly outperforms other schemes in the literature in terms of bit error rate and system throughput.