Zero-Error Relaying for Primitive Relay Channels

In a primitive relay channel, a new one-shot relaying scheme termed color-and-forward is proposed that guarantees a probability of error equal to zero. This relaying scheme constructs a relaying compression graph of relay outputs based on the joint conditional distribution of the relay and destination outputs, and forwards a minimum coloring of this graph. The n-letter extension of the proposed color-and-forward scheme is shown to be optimal in the sense that it results in the smallest needed out-of-band relay to destination link rate for the overall message rate to equal the single-input multiple-output outer bound for any fixed number of channel uses. This is used to obtain an upper bound on the asymptotic minimal relay to destination link rate needed to achieve the single-input multiple-output outer bound.