The collision channel without feedback

Abstract

A model is proposed for the situation where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> users share a common communication resource but, because of unknown time offsets among their clocks, cannot transmit their data packets in a time-sharing mode and, because of the lack of a feedback link, can never determine these time offsets and also can never be sure of the outcomes of their individual packet transmissions. Each user is required to make his packet transmissions at times determined by a protocol signal that is independent of the data to be sent. The capacity and zero-error capacity regions of this channel are determined for both the unsynchronized and slot-synchronized cases; these four regions are shown to coincide. It is further shown that a dense set of rate points on the outer boundary of this region can be achieved in the slot-synchronized case. Specific constructions of protocol sequences for achieving these points are given, and the technique of "decimation decoding" is introduced for identifying the sender of each successfully transmitted packet. Maximum-erasure burst-correcting codes over an alphabet of arbitrary size are constructed and shown to suffice for reconstructing the packets lost in "collisions" when these protocol sequences are used.