The capacity regions of some classes of deterministic relay channels

The capacity regions of two new classes of deterministic relay channels are established. In the first class of deterministic relay channels, the family of conditional probability distributions describing the relay channel can be written as p(y
₂
, y
₃
|x
₁
, x
₂
)=p(y
₃
|x
₁
, x
₂
)p(y
₂
|s, x
₂
, y
₃
) where s is a deterministic function of x
₁
, i.e., s=f
₁
(x
₁
). In addition, we require that Srarr(X
₂
, Y
₂
)rarrY
₃
form a Markov chain for all input probability distributions p(x
₁
, x
₂
). In the second class of deterministic relay channels, there is causal noiseless feedback from relay to sender and the relay output is a deterministic function of x
₁
, x
₂
, and y
₃
, i.e., y
₂
=f
₃
(x
₂
, x
₂
, y
₃
). We consider two alternative schemes to achieve the capacity. The first is based on a generalized strategy of Gabbai and Bross. The second strategy is based on a ldquohash-and-forwardrdquo scheme by Cover and Kim.