Capacity of a Class of Deterministic Relay Channels

The capacity of a class of deterministic relay channels with the transmitter input X, the receiver output Y, the relay output Y
₁
= f(X,Y), and a separate communication link from the relay to the receiver with capacity R
₀
, is shown to be C(R
₀
) =max min{I(X;Y) + R
₀
, I(X;Y,Y
₁
)}. p(x) Thus every bit from the relay is worth exactly one bit to the receiver. Two alternative coding schemes are presented that achieve this capacity. The first scheme, "hash-and-forward", is based on a variation of the usual random binning on the relay outputs, while the second scheme uses the usual "compress-and- forward". In fact, these two schemes can be combined to give a class of optimal coding schemes. As a corollary, this relay capacity result confirms a conjecture by Ahlswede and Han on the capacity of a channel with rate-limited state information at the decoder in the special case when the channel state is recoverable from the channel input and output.