Delay-Complexity Trade-Off of Random Linear Network Coding in Wireless Broadcast

In wireless broadcast, random linear network coding (RLNC) over GF( $2^{L}$ ) is known to asymptotically achieve the optimal completion delay with increasing $L$. However, the high decoding complexity hinders the potential applicability of RLNC schemes over large GF( $2^{L}$ ). In this paper, a comprehensive analysis of completion delay and decoding complexity is conducted for field-based systematic RLNC schemes in wireless broadcast. In particular, we prove that the RLNC scheme over GF(2) can also asymptotically approach the optimal completion delay per packet when the packet number goes to infinity. Moreover, we introduce a new method, based on circular-shift operations, to design RLNC schemes which avoid multiplications over large GF( $2^{L}$ ). Based on both theoretical and numerical analyses, the new RLNC schemes turn out to have a much better trade-off between completion delay and decoding complexity. In particular, numerical results demonstrate that the proposed schemes can attain average completion delay just within 5% higher than the optimal one, while the decoding complexity is only about 3 times the one of the RLNC scheme over GF(2).