Optimality of separate network-channel coding for three messages

We study the optimality of separate network-channel coding (SJNC) for three independent sources on the orthogonal access scheme. By SJNC, we mean that the network coded messages are formed from the binary XOR of the sources´ messages, followed by channel encoding which is independent of the network coding. Decoding is performed jointly across the network and channel codes. Using standard random coding and joint typical set decoding, we obtain an achievable rate region for SJNC. By allowing time-sharing of codes, the rate region coincides with the capacity region of joint network-channel coding, where network and channel coding are performed jointly. This proves, surprising, that there is no loss of optimality for SJNC.