New constructions of cyclic subspace codes

TL;DR

This paper introduces two new methods for constructing cyclic subspace codes using Sidon spaces, resulting in larger codes with optimal properties and approaching theoretical bounds.

Contribution

The paper presents novel constructions of Sidon spaces that generate larger cyclic subspace codes with optimal minimum distance and near-optimal size.

Findings

Constructed cyclic subspace codes with minimum distance 2k-2.

Codes have sizes larger than previous literature.

Size approaches sphere-packing bound as parameters grow.

Abstract

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.