A relaying graph and special strong product for zero-error problems in primitive relay channels

A primitive relay channel (PRC) has one source(S) communicating a message to one destination (D) with thehelp of a relay (R). The link between R and D is consideredto be noiseless, of finite capacity, and parallel to the linkbetween S and (R,D). Prior work has established, for any fixednumber of channel uses, the minimal R-D link rate needed sothat the overall S-D message rate equals the zero-error single-input multiple output outer bound (Problem 1). The zero-errorrelaying scheme was expressed as a coloring of a carefully defined“relaying compression graph”. It is shown here that this relayingcompression graph for n channel uses is not obtained as a strongproduct from its n = 1 instance. Here we define a new graph, the“primitive relaying graph” and a new “special strong product”such that the n-channel use primitive relaying graph correspondsto the n-fold special strong product of the n = 1 graph. We showhow the solution to Problem 1 can be obtained from this newprimitive relaying graph directly. Further study of this primitiverelaying graph has the potential to highlight the structure ofoptimal codes for zero-error relaying.