In wireless broadcast, random linear network coding (RLNC) over GF(2L) 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).

中文翻译：

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无线广播中随机线性网络编码的延迟复杂度权衡


在无线广播中，已知GF（2L）上的随机线性网络编码（RLNC）随着L的增加而渐近地实现最佳完成延迟。然而，高解码复杂度阻碍了RLNC方案在大GF（2L）上的潜在适用性。本文对无线广播中基于现场的系统RLNC方案的完成延迟和解码复杂度进行了综合分析。特别是，我们证明了当数据包数量趋于无穷大时，基于 GF(2) 的 RLNC 方案也可以渐近地接近每个数据包的最佳完成延迟。此外，我们引入了一种基于循环移位运算的新方法来设计 RLNC 方案，该方案避免了大 GF(2 L ) 上的乘法。基于理论和数值分析，新的 RLNC 方案在完成延迟和解码复杂度之间具有更好的权衡。特别是，数值结果表明，所提出的方案的平均完成延迟仅比最佳方案高出 5% 以内，而解码复杂度仅为 GF(2) 的 RLNC 方案的约 3 倍。