We consider a situation where several users send information over a single additive channel. We suppose the transmission is synchronized and that at most m of the total number of potential T users transmit simultaneously. The receiver gets a message which is the sum of the messages of the active users at that moment. In order to recover the active users we have to be sure that all possible combinations of up to m users produce distinct messages at the receiver end. We require the received messages to be as far from each other as possible.

The above situation can be described in terms of superimposed codes. In this thesis we study superimposed codes in the binary Hamming spaces and in Euclidean spaces. These spaces arc metric spaces. The distance between the elements in the former is defined as the number of positions where they differ. In the latter the usual Euclidean distance provides the metric. Both binary and Euclidean superimposed codes have four parameters: length, strength, minimum distance and cardinality.

We investigate the best cardinality that a superimposed code can achieve for a given set of the other three parameters. Lower bounds on this quantity are provided by a number of new constructions. In many cases these bounds improve upon any previously known lower bound. Upper bounds on the best cardinality arc described for additive superimposed codes in the binary Hamming spaces. We also show the optimality of a sequence of codes constructed with help of Hadamard matrices. A conjecture on the classification of the so called doubly perfect superimposed codes is proved.

The simplex codes in the Euclidean spaces provide the best constellations under many different criteria. For superimposed codes, however, we show that the simplex codes are normally not the best.