New constructions of large cyclic subspace codes via Sidon spaces

Abstract

<p style='text-indent:20px;'>Subspace codes, especially cyclic subspace codes, have attracted wide attention due to their applications in random network coding. In [<xref ref-type="bibr" rid="b14">14</xref>], 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 <inline-formula><tex-math id="M1">\begin{document}$ l{q^k}\left( {\left\lceil {\frac{n}{{4k}}} \right\rceil - 1} \right)\frac{{{q^n} - 1}}{{q - 1}} $\end{document}</tex-math></inline-formula> and minimum distance <inline-formula><tex-math id="M2">\begin{document}$ 2k-2 $\end{document}</tex-math></inline-formula> is further given which improves and generalizes the previously known constructions, where <inline-formula><tex-math id="M3">\begin{document}$ n, k, l $\end{document}</tex-math></inline-formula> are positive integers, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> is a multiple of <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ l\leq k $\end{document}</tex-math></inline-formula>. Furthermore, in the case <inline-formula><tex-math id="M7">\begin{document}$ n = 3k, $\end{document}</tex-math></inline-formula> by considering the orbits of distinct Sidon spaces and the orbit of <inline-formula><tex-math id="M8">\begin{document}$ {\mathbb{F}_{{q^k}}}, $\end{document}</tex-math></inline-formula> a cyclic subspace code with size <inline-formula><tex-math id="M9">\begin{document}$ 2({q^k}-1) \frac{{{q^n} - 1}}{{q - 1}} + {q^{2k}}+{q^k} + 1 $\end{document}</tex-math></inline-formula> and minimum distance <inline-formula><tex-math id="M10">\begin{document}$ 2k-2 $\end{document}</tex-math></inline-formula> 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.

Topics & Concepts