Code constructions for asynchronous random multiple-access to the adder channel

A situation where up to T randomly chosen users, out of M potential ones, simultaneously transmit data over the noiseless adder channel (AC) is considered. These T (or fewer) active users operate independently, and because of unknown time offsets among their clocks and different delays that the various messages incur during transmission, both block and bit synchronism are precluded. Code constructions that ensure error-free asynchronous communication over the adder channel are presented. The information rate of these codes=approaches 1/T bits per user. If exactly T users are active, use of these codes leads to an aggregate rate arbitrarily close to 1 bit per channel use. Furthermore, if the occurrence of simultaneous transmissions of more than T users is made known to the active transmitters, such an aggregate rate can be maintained for arbitrary T, T⩽M by means of an adaptive use of the codes