  A subspace of a finite field is called a Sidon space if the product of any
two of its nonzero elements is unique up to a scalar multiplier from the base
field. Sidon spaces, introduced by Roth et al. (IEEE Trans Inf Theory 64(6):
4412-4422, 2018), have a close connection with optimal full-length orbit codes.
In this paper, we present two constructions of Sidon spaces. The union of Sidon
spaces from the first construction yields cyclic subspace codes in
$\mathcal{G}_{q}(n,k)$ with minimum distance $2k-2$ and size $r(\lceil
\frac{n}{2rk} \rceil
-1)((q^{k}-1)^{r}(q^{n}-1)+\frac{(q^{k}-1)^{r-1}(q^{n}-1)}{q-1})$, where $k|n$,
$r\geq 2$ and $n\geq (2r+1)k$, $\mathcal{G}_{q}(n,k)$ is the set of all
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. The union of Sidon spaces
from the second construction gives cyclic subspace codes in
$\mathcal{G}_{q}(n,k)$ with minimum distance $2k-2$ and size $\lfloor
\frac{(r-1)(q^{k}-2)(q^{k}-1)^{r-1}(q^{n}-1)}{2}\rfloor$ where $n= 2rk$ and
$r\geq 2$. Our cyclic subspace codes have larger sizes than those in the
literature, in particular, in the case of $n=4k$, the size of our resulting
code is within a factor of $\frac{1}{2}+o_{k}(1)$ of the sphere-packing bound
as $k$ goes to infinity.
