A Precoding-Based Compute-and-Forward

The compute-and-forward scheme can achieve high rates with low complexity since it directly decodes an integer linear combination of messages instead of decoding all messages. However, the number of degrees of freedom (DoF) of this approach is 2/(1+1/K), where K is the number of transmitters. We propose an improved scheme by moving the scaling factors from the receiver in the conventional method to the transmitters to overcome the Diophantine tradeoff. First, we optimize the scalars, also called the precoders, under the transmitting power constraint, to obtain the maximum sum rate. Then we show that this proposed scheme attains a DoF of K/2 + K/[2(K-1)] since it avoids amplifying the additive noise. Moreover, we present an iterative algorithm to find the optimal precoder and the integer matrix of the effective channel.