Practical compute-and-forward approaches for the multiple access relay channel

We consider a multiple access relay channel (MARC) network consisting of two sources, one relay, and one common destination applying compute-and-forward (CF) strategy. We show that the direct application of CF to the MARC network results in poor error performance bounded by (p + 1)
^-1
, the probability of rank deficiency of the coefficient matrix over F
_p
. To solve this problem, we propose two practical approaches. First, given an optimal coefficient vector at the relay, the destination is restricted to select a coefficient vector ensuring a full rank coefficient matrix. Second, given an optimal coefficient vector at the destination obtained via a small amount of feedback, the relay is restricted to choose a coefficient vector guaranteeing a full rank coefficient matrix. We simulate these CF implementation strategies using self-similar nested E
₈
lattice codes and confirm that both of the proposed schemes outperform the direct implementation in terms of achievable transmission rate and frame-error-rate performance. Furthermore, we confirm that with a small amount of feedback, the second strategy is better than the first one. In addition, we present in detail a modified Fincke-Pohst algorithm for computing the coefficient candidates and show its efficiency compared to an exhaustive search.