  In this paper, we propose different practical distributed schemes to solve
the rank failure problem in the compute and forward (CMF)-based multi-user
multi-relay networks without central coordinator, in which the relays have no
prior information about each other. First, a new relaying strategy based on
CMF, named incremental compute-and-forward (ICMF), is proposed that performs
quite well in terms of the outage probability. We show that the distributed
ICMF scheme can even outperform the achievable rate of centralized optimal CMF
in strong enough inter relay links, with much less complexity. Then, as the
second scheme, amplify-forward and compute (AFC) is introduced in which the
equations are recovered in the destination rather than in the relays. Finally,
ICMF and AFC schemes are combined to present hybrid compute-amplify and forward
(HCAF) relaying scheme, which takes advantages of both ICMF, and AFC and
improves the performance of the ICMF considerably. We evaluate the performance
of the proposed strategies in terms of the outage probability and compare the
results with those of the conventional CMF strategy, the Decode and Forward
(DF) strategy, and also the centralized optimal CMF. The results indicate the
substantial superiority of the proposed schemes compared with the conventional
schemes, specially for high number of users and relays.
