Simultaneous communication in noisy channels

A sender wishes to broadcast a message of length \(n\) over an alphabet to \(r\) users, where each user \(i\), \(1 \leq i \leq r\) should be able to receive one of \(m_i\) possible messages. The broadcast channel has noise for each of the users (possibly different noise for different users), who cannot distinguish between some pairs of letters. The vector \((m_1, m_2,...s, m_r)_{(n)}\) is said to be feasible if length \(n\) encoding and decoding schemes exist enabling every user to decode his message. A rate vector \((R_1, R_2,..., R_r)\) is feasible if there exists a sequence of feasible vectors \((m_1, m_2,..., m_r)_{(n)}\) such that \(R_i = \lim_{n \mapsto \infty} \frac {\log_2 m_i} {n}, {for all} i\). We determine the feasible rate vectors for several different scenarios and investigate some of their properties. An interesting case discussed is when one user can only distinguish between all the letters in a subset of the alphabet. Tight restrictions on the feasible rate vectors for some specific noise types for the other users are provided. The simplest non-trivial cases of two users and alphabet of size three are fully characterized. To this end a more general previously known result, to which we sketch an alternative proof, is used. This problem generalizes the study of the Shannon capacity of a graph, by considering more than a single user.