A class of codes for a T active users out of N multiple-access communication system

A code construction for a T active users out of N multiple-access system (TANMAS) is discussed. The multiple-access channel (MAC) that is used in the TANMAS is a discrete-time noiseless real adder channel used without feedback with N real or binary inputs. Each input may be affected by an unknown, slowly varying channel gain and by an unknown, slowly varying channel offset. A set of N codes is constructed such that the sum of codewords of any known set of T or less active users is uniquely decodable. The sum rate of the codes approaches one from above if the input alphabet of the MAC is binary, and it approaches one from below if the input symbols are real numbers. For the case when the set of active users is unknown. For the case when the set of active users is unknown, it is shown that this set can be identified uniquely (in addition to unique decodability) provided that at most T/2 users are active simultaneously. The sum rate is then reduced to 1/2, which is approached from above and from below, respectively, for binary and for real channel inputs. A simple decoding algorithm which operates over the reals is given for the case when the set of active users is known. It is pointed out that one of the applications of the given codes is in hybrid multiple-access systems that use both multiple-access coding and collision resolution