The capacity of digital links in tandem

Suppose we have
L
digital links in tandem. Between successive links we have a repeater. The links are each white Gaussian channels with no bandwidth constraint. The repeater may either be a {\em binary repeater}, which sends on each bit separately having made a binary decision on each, or a {\em Shannon repeater}, which perfectly reproduces at one end the bits that were transmitted at the other end of the link. What is the loss in capacity if we use only binary repeaters and code for the entire L links at the transmit end? For large
L
, the capacity drops by a factor asymptotic to In
L
, and the normalized optimum time
T
per symbol
T_{\opt}^{L}
is also asymptotic to
\ln L
. Arbitrarily short
L
gives
0
capacity for
L > 1
. More precise asymptotic results are obtained and compared with computed numerical values. These results show when it pays to code each link separately in digital data transmission.