Combinatorial structure and capacity of the permuting relay channel

Blackwell's trap-door channel is an interesting example of a finite state channel. Its deterministic version, that is, the permuting channel, has been studied by Ahlswede and Kaspi ina multiterminal information-theoretic framework. They determined the capacities of permuting jammer channels and relay channels for some special cases. The capacity problem for permuting relay channels is completely solved. More specifically, when a is the cardinality of alphabet, and
\beta
is the number of available storage locations in the channel, the capacity
C_{R}(\alpha, \beta)
of the permuting relay channel is given by
\log \lambda
, where
\lambda
denotes the maximum eigenvalue of a matrix
Q
derived from the state-transition mechanism associated with the channel.