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 $p_0=\max \{i\in \mathbb{N}^+: \lfloor \frac{r}{i}\rfloor>\lfloor \frac{r}{i+1} \rfloor \}$, the first construction produces a cyclic CDC in $\mathcal{G}_q(n, k)$ with minimum distance $2k-2$ and size $\frac{\left((r+\sum\limits_{i=2}^{p_0}(\lfloor \frac{r}{i}\rfloor-\lfloor \frac{r}{i+1} \rfloor))(q^k-1)(q-1)+r\right)(q^k-1)^{r-1}(q^n-1)}{q-1}$. 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 $\frac{\left((r-1+\sum\limits_{i=2}^{p_0}(\lceil \frac{r}{i}\rceil-\lfloor \frac{r}{i+1} \rfloor-1))(q^k-1)(q-1)+r-1\right)(q^k-1)^{r-2}\lfloor \frac{q^k-2}{2}\rfloor(q^n-1)}{q-1}$ 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.