  A general time-varying feedback coding scheme is proposed for $M$-user fully
connected symmetric Gaussian interference channels. Based on the analysis of
the general coding scheme, we prove a theorem which gives a criterion for
designing good time-varying feedback codes for Gaussian interference channels.
The proposed scheme improves the Suh-Tse and Kramer inner bounds of the channel
capacity for the cases of weak and not very strong interference when $M=2$.
This capacity improvement is more significant when the signal-to-noise ratio
(SNR) is not very high. In addition, our coding scheme can be proved
mathematically and numerically to outperform the Kramer code for $M\geq 2$ when
Signal to Noise Ratio (SNR) is equal to Interference to Noise Ratio (INR).
Besides, the generalized degrees-of-freedom (GDoF) of our proposed coding
scheme can be proved to be optimal in the all network situations (very weak,
weak, strong, very strong) for any $M$. The numerical results show that our
coding scheme can attain better performance than the Suh-Tse coding scheme for
$M=2$ or the Mohajer-Tandon-Poor lattice coding scheme for $M>2$. Furthermore,
the simplicity of the encoding/decoding algorithms is another strong point of
our proposed coding scheme compared with the Suh-Tse coding scheme when $M=2$
and the Mohajer-Tandon-Poor lattice coding scheme when $M>2$. More importantly,
our results show that an optimal coding scheme for the symmetric Gaussian
interference channels with feedback can be achieved by only using marginal
posterior distributions under a better cooperation strategy between
transmitters.
