Upper Bound for Uniquely Decodable Codes in a Binary Input N-User Adder Channel

The binary input N-user adder channel models a communicant ion media accessed simultaneously by N users. In this model each user transmits binary sequences and the channel's output on each bit slot equals the sum of the corresponding N inputs. A uniquely decodable code for this channel is a set of N codes - a code for each of the N users - such that the receiver can determine all possible combinations of transmitted codewords from their sum. Van-Tilborg presented a method for determining an upper bound on the size of a uniquely decodable code for the two-user binary adder channel. He showed that for sufficiently large block length this combinatorial bound converges to the corresponding capacity region boundary. In the present work we use a similar method to derive an upper bound on the size of a uniquely decodable code for the binary input N-user adder channel. The new combinatorial bound is iterative - i.e., the bound for the (N - I)-user case can be obtained by projecting the N-user bound on (N - 1) combinatorial variables and in particular it subsumes the two-user result. For sufficiently large block length the N-user bound converges to the capacity region boundary of the binary input N-user adder channel.