Existence of large cyclic subspace codes in every dimension

Let $\mathcal{G}_{q}(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, and let a cyclic code be a subspace code invariant under multiplication by every element of $\mathbb{F}_{q^n}^{*}$. The size of a code is its number of codewords, and its minimum distance is the minimum subspace distance between distinct codewords. Cyclic subspace-code existence conjecture. For every positive integers $n,k$ such that $k\le n/2$, there exists a cyclic code of size

$$\frac{q^n-1}{q-1}$$

in $\mathcal{G}_{q}(n,k)$ and minimum distance $2k-2$. This conjecture concerns the existence of large cyclic subspace codes with prescribed dimension and distance; the paper presents general construction techniques because existing methods fall short for cyclic codes, while the cited prior work provides the conjectural formulation.