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.

翻译：一个有限域的子空间称为Sidon空间，若其中任意两个非零元素的乘积在基域标量乘意义下唯一。Roth等人（IEEE Trans Inf Theory 64(6): 4412-4422, 2018）引入的Sidon空间与最优全长轨道码有密切联系。本文给出Sidon空间的两种构造。第一种构造中Sidon空间的并集生成$\mathcal{G}_{q}(n,k)$中的循环子空间码，其最小距离为$2k-2$，大小为$r(\lceil \frac{n}{2rk} \rceil -1)((q^{k}-1)^{r}(q^{n}-1)+\frac{(q^{k}-1)^{r-1}(q^{n}-1)}{q-1})$，其中$k|n$，$r\geq 2$，$n\geq (2r+1)k$，$\mathcal{G}_{q}(n,k)$是$\mathbb{F}_{q}^{n}$中所有$k$维子空间构成的集合。第二种构造中Sidon空间的并集生成$\mathcal{G}_{q}(n,k)$中的循环子空间码，其最小距离为$2k-2$，大小为$\lfloor \frac{(r-1)(q^{k}-2)(q^{k}-1)^{r-1}(q^{n}-1)}{2}\rfloor$，其中$n= 2rk$且$r\geq 2$。本文构造的循环子空间码比现有文献中的码具有更大尺寸，特别地，当$n=4k$时，所得码的尺寸在$k$趋于无穷时与球填充界相差不超过$\frac{1}{2}+o_{k}(1)$因子。