  We consider cooperative relay communication in a fading channel environment
under the Orthogonal Amplify and Forward (OAF) and Orthogonal and
Non-Orthogonal Selection Decode and Forward (OSDF and NSDF) protocols. For all
these protocols, we compute the Diversity-Multiplexing Gain Tradeoff (DMT). We
construct DMT optimal codes for the protocols which are sphere decodable and,
in certain cases, incur minimum possible delay. Our results establish that the
DMT of the OAF protocol is identical to the DMT of the Non-Orthogonal Amplify
and Forward (NAF) protocol. Two variants of the NSDF protocol are considered:
fixed-NSDF and variable-NSDF protocol. In the variable-NSDF protocol, the
fraction of time duration for which the source alone transmits is allowed to
vary with the rate of communication. Among the class of static
amplify-and-forward and decode-and-forward protocols, the variable-NSDF
protocol is shown to have the best known DMT for any number of relays apart
from the two-relay case. When there are two relays, the variable-NSDF protocol
is shown to improve on the DMT of the best previously-known protocol for higher
values of the multiplexing gain. Our results also establish that the fixed-NSDF
protocol has a better DMT than the NAF protocol for any number of relays.
Finally, we present a DMT optimal code construction for the NAF protocol.
