  In this work, we explore the introduction of distributed space-time codes in
decode-and-forward (DF) protocols. A first protocol named the Asymmetric DF is
presented. It is based on two phases of different lengths, defined so that
signals can be fully decoded at relays. This strategy brings full diversity but
the symbol rate is not optimal. To solve this problem a second protocol named
the Incomplete DF is defined. It is based on an incomplete decoding at the
relays reducing the length of the first phase. This last strategy brings both
full diversity and full symbol rate. The outage probability and the simulation
results show that the Incomplete DF has better performance than any existing DF
protocol and than the non-orthogonal amplify-and-forward (NAF) strategy using
the same space-time codes. Moreover the diversity-multiplexing gain tradeoff
(DMT) of this new DF protocol is proven to be the same as the one of the NAF.
