Full-Diversity Binary Frame-Wise Network Coding for Multiple-Source Multiple-Relay Networks Over Slow-Fading Channels

We study the design of network codes for -source -relay wireless networks over slow-fading channels. Binary frame-wise network coding (BFNC) based on cyclic-shifting matrices is developed to achieve full diversity and good coding gain. We develop a criterion in the context of BFNC that if satisfied guarantees to achieve full diversity gain. Based on this criterion, we propose an algorithm to design low-complexity encoders for a BFNC scheme by exploiting quasi-cyclic low-density parity-check (LDPC) code structures. We also design practical decoders based on the belief propagation decoding principle with a focus on large block lengths. Numerical results demonstrate that our BFNC schemes have substantial benefits over previous complex field and Galois field network coding schemes in the sense that our BFNC schemes can achieve full diversity gain and high coding gain for arbitrary block lengths with low encoding/decoding complexity.