. The complexity is $O(L\sqrt{P\norm{\h}^2})$ for a given channel vector $\h\in\Rbb^L$ and signal power constraint $P$, and is of average value $O(P^{0.5}L^{1.5})$ for i.i.d. standard Gaussian channel entries.

-   For the complex-valued channels, we demonstrate how to apply our method in an efficient way to find the complex-valued coefficient vector.

-   Extensive simulation results are presented to compare the effectiveness and efficiency of our method with the existing methods.

Part of this work has been presented in [@Zhou2014]. One main improvement here is the complexity order for i.i.d. Gaussian channel entries is further reduced from 3 to 1.5.

In the following, we will first introduce the system model of AWGN networks as well as the CF network coding design problem in Section \[section:ProblemStatement\]. Then in Section \[section:ProposedMethod\], we will present our proposed method in detail. Numerical results will be shown in Section \[section:NumericalResults\]. Finally, we will conclude our work in Section \[section:Conclusions\].

[*Notation.*]{} Let $\Rbb$ be the real field, $\Cbb$ be the complex field, and $\Zbb$ be the ring of integers. Boldface lowercase letters denote column vectors, and boldface uppercase letters denote matrices, e.g., $\w\in\Rbb^L$ and $\W\in\Rbb^{M\times L}$. $\norm{\w}$ denotes the $\ell^2$-norm of $\w$, and $\w^T$ denotes the transpose of $\w$. For a vector $\w$, let $\w(\ell)$ be the element with index $\ell$, and $\w(i\!:\!j)$ be the vector composed of elements with indices from $i$ to $j$. For a matrix $\W$, let be the submatrix containing elements with row indices from $i$ to $j$ and column indices from $k$ to $\ell$, be the submatrix containing elements with row indices from $i$ to $j$ and column index $k$, $\W(i,k\!:\!\ell)$ be the submatrix containing elements with row index $i$ and column indices from $k$ to $\ell$, and $\W(i,j)$ be the element with row index $i$ and column index $j$. Let $\floor{x}$ and $\ceil{x}$, i.e., the corresponding floor and ceiling f