Binary Field Network Coding Design for Multiple-Source Multiple-Relay Networks

We study the design of network codes for M-source, N-relay wireless networks over slow fading channels. Specifically, vector-wise binary field network coding (BFNC) schemes are proposed. In the construction of our BFNC schemes, we utilize a diversity achieving criterion which can be expressed in terms of the linear independence of quasi-cyclic matrices. Our codes can be implemented with low-complexity encoders at the relays as only binary operations are used. Meanwhile at the destination, for small code lengths, ML decoding can be applied. For large code lengths, we propose a modified BP decoder with low decoding complexity. From analysis and simulations, we show that our proposed BFNC schemes can achieve full diversity for the ML decoder, as well as full diversity for the modified BP decoder we propose for large block lengths. Our simulations also show that our proposed BFNC schemes achieve a higher coding gain relative to previous network coding schemes.