Subspaces coding plays an important role in error correction of random network coding. To study their properties and find good constructions, the notion of cyclic subspace codes was introduced using the extension field structure of the ambient space. Those cyclic constant-dimension subspace codes (CDCs) with optimal minimum distances may have additional structures that could be effectively applied in encoding and decoding algorithms (see Trautmann et al., IEEE Trans Inf Theory 59(11):7386–7404, 2013; Etzion and Vardy, IEEE Trans Inf Theory 57(2):1165–1173, 2011; Braun et al., Forum Math Pi 4(e7):1–14, 2016; Kohnert and Kurz, Math Methods Comput Sci 5393:31–42, 2008). In this paper, two new constructions of Sidon spaces are given by tactfully adding new parameters and flexibly varying the number of parameters. Under the parameters \( n= (2r+1)k, r \ge 2 \) and \(p_0=\max \{i\in \mathbb {N}^+: \lfloor \frac{r}{i}\rfloor >\lfloor \frac{r}{i+1} \rfloor \}\) , the first construction produces a cyclic CDC in \(\mathcal {G}_q(n, k)\) with minimum distance \(2k-2\) and size \(\frac{\left( (r+\sum _{i=2}^{p_0}(\lfloor \frac{r}{i}\rfloor -\lfloor \frac{r}{i+1} \rfloor ))(q^k-1)(q-1)+r\right) (q^k-1)^{r-1}(q^n-1)}{q-1}\) . Given parameters \(n=2rk,r\ge 2\) and if \(r=2\) , \(p_0=1\) , otherwise, \(p_0=\max \{ i\in \mathbb {N}^+: \lceil \frac{r}{i}\rceil -1>\lfloor \frac{r}{i+1} \rfloor \}\) , a cyclic CDC in \(\mathcal {G}_q(n, k)\) with minimum distance \(2k-2\) and size \(\frac{\left( (r-1+\sum _{i=2}^{p_0}(\lceil \frac{r}{i}\rceil -\lfloor \frac{r}{i+1} \rfloor -1))(q^k-1)(q-1)+r-1\right) (q^k-1)^{r-2}\lfloor \frac{q^k-2}{2}\rfloor (q^n-1)}{q-1}\) is produced by the second construction. The sizes of our cyclic CDCs are larger than the best known results. In particular, in the case of \(n=4k\) , when k goes to infinity, the ratio between the size of our cyclic CDC and the Sphere-packing bound (Johnson bound) is approximately equal to \(\frac{1}{2}\) . Moreover, for a prime power q and positive integers k, s with \(1\le s< k-1\) , a cyclic CDC in \(\mathcal {G}_q(N, k)\) of size \(e\frac{q^N-1}{q-1}\) and minimum distance \(\ge 2k-2s\) is provided by subspace polynomials, where N, e are positive integers. Our construction generalizes previous results and, under certain parameters, provides cyclic CDCs with larger sizes or more admissible values of N than constructions based on trinomials.
