*. A GQC code in ${\mathcal R}$ is also a left $R$-submodule of ${\mathcal R}$, and we call it a *module skew GQC code*. Most of our results on skew cyclic codes and skew GQC codes in this paper depend on the fact that $x^m-1$ is a central element of $R$. Since in the module skew case this is not ture anymore, some results stated in this paper cannot be held. Therefore, the structural properties of module skew cyclic and module skew GQC codes are also interesting open problems for further consideration. Another interesting open problem is to find some new or good linear codes over finite fields from skew GQC codes.

[**Acknowledgments**]{} This research is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), the National Natural Science Foundation of China (Nos. 61171082, 60872025, 10990011).

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