New constructions of large cyclic subspace codes via Sidon spaces

Subspace codes, especially cyclic subspace codes, have attracted wide attention due to their applications in random network coding. In [14], Roth et al. presented the idea that cyclic subspace codes can be constructed employing Sidon spaces. In this paper, several kinds of Sidon spaces are first constructed and a cyclic subspace code with size \begin{document}$ l{q^k}\left( {\left\lceil {\frac{n}{{4k}}} \right\rceil - 1} \right)\frac{{{q^n} - 1}}{{q - 1}} $\end{document} and minimum distance \begin{document}$ 2k-2 $\end{document} is further given which improves and generalizes the previously known constructions, where \begin{document}$ n, k, l $\end{document} are positive integers, \begin{document}$ n $\end{document} is a multiple of \begin{document}$ k $\end{document} and \begin{document}$ l\leq k $\end{document}. Furthermore, in the case \begin{document}$ n = 3k, $\end{document} by considering the orbits of distinct Sidon spaces and the orbit of \begin{document}$ {\mathbb{F}_{{q^k}}}, $\end{document} a cyclic subspace code with size \begin{document}$ 2({q^k}-1) \frac{{{q^n} - 1}}{{q - 1}} + {q^{2k}}+{q^k} + 1 $\end{document} and minimum distance \begin{document}$ 2k-2 $\end{document} is obtained. As a consequence, we obtain more cyclic subspace codes with larger size of codewords than the previous works without decreasing the minimum distance.