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.