Coding scheme is discussed for M-Choose-T communication in which at most T active users out of M potential users simultaneously transmit their messages over a common channel. The multiple-access channel considered in this paper is assumed to be a time-discrete noiseless adder channel without feedback with T binary inputs and one real-valued output, and is used on the assumption of perfect block and bit synchronization among users. In this paper a new class of uniquely decodable codes is proposed in order to realize error-free M-Choose-T communication over the adder channel described above. These codes are uniquely decodable in the sense that not only the set of active users can be specified but also their transmitted messages can be recovered uniquely as long as T or fewer users are active simultaneously. It is shown that these codes have a simple decoding algorithm and can achieve a very high sum rate arbitrarily close to unity if exactly T users are active.