Capacity region of the reciprocal deterministic 3-way channel via Δ-Y transformation

A linear shift deterministic 3-way channel with reciprocal channel gains is considered in this work. The 3-way channel is an extension of the 2-way channel introduced by Shannon. Here, a number of six messages is exchanged, one message from each user to the two other users. Each user operates in a full-duplex mode. We derive the capacity region of this 3-way channel w.r.t. the linear shift deterministic channel model. To this end, first, an outer bound is derived using cutset and genie-aided upper bounds. Then, it is noted that the outer bound bears a resemblance to the capacity region of a related linear shift deterministic Y-channel. Utilizing a Δ-Y transformation, the optimal scheme for the related Y- channel is modified in a way that achieves the outer bound of the 3-way channel. Mainly, the capacity achieving communication schemes are based on multi-way relaying by signal alignment, interference neutralization and backward decoding. We also consider a scheme which is based on interference alignment only. It turns out, that for the symmetric linear deterministic 3-way channel, this scheme is optimal. Thus, backward decoding and the resulting delays are avoided.