New constructions of cyclic constant-dimension subspace codes based on Sidon spaces and subspace polynomials

Abstract: 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 $n= (2r+1)k, r \ge2$ and $p_0=\max {i\in \mathbb{N}^+: \lfloor \frac{r}{i}\rfloor&gt;\lfloor \frac{r}{i+1} \rfloor }$, the first construction produces a cyclic CDC in $\mathcal{G}<em>q(n, k)$ with minimum distance $2k-2$ and size $$\frac{\left((r+\sum\limits</em>{i=2}^{{p_0}(\lfloor} \frac{r}{i}\rfloor-\lfloor \frac{r}{i+1} \rfloor))(q<sup>k-1)(q-1)+r\right)(q<sup>k-1)<sup>{r-1}(q<sup>n-1)}{q-1}$$. Given parameters $n=2rk,r\ge 2$ and if $r=2$, $p_0=1$, otherwise, $p_0=\max{ i\in \mathbb{N}^+: \lceil\frac{r}{i}\rceil-1&gt;\lfloor \frac{r}{i+1} \rfloor }$, a cyclic CDC in $\mathcal{G}<em>q(n, k)$ with minimum distance $2k-2$ and size $$\frac{\left((r-1+\sum\limits</em>{i=2}^{p_0}(\lceil \frac{r}{i}\rceil-\lfloor \frac{r}{i+1} \rfloor-1))(q^{k-1)(q-1)+r-1\right)(q^{k-1)^{r-2}\lfloor}} \frac{q<sup>k-2}{2}\rfloor(q<sup>n-1)}{q-1}$$ 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 $1\le s&lt; k-1$, a cyclic CDC in $\mathcal{G}_q(N, k)$ of size $e\frac{q<sup>N-1}{q-1}$ 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.