Superimposed codes in the Hamming space

We consider the problem of simultaneous identification and transmission of information by means of superimposed codes. It is assumed that a total population of T users share a common channel. At each time at most in of the T users are active. The output from the channel is the modulo-2 sum of the binary codewords transmitted by the active users, possibly disturbed by errors. A superimposed code is actually a set of codes-one for each user-designed in such a way that both the (at most) m active users and their messages can be identified, even in the presence of a permissible number of errors in the channel. We give several explicit constructions for superimposed codes and analyze in detail their performance. In several of the cases under study we offer codes that are proved to be best possible.<
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