Deep-hole classification conjecture for projective Reed–Solomon codes

Let $q$ be odd and $2\le k\le q-2$. Represent a word of $\mathbb{F}_q^{q+1}$ as $(u_f,v)$, where $u_f=(f(\alpha_1),\ldots,f(\alpha_q))$ is defined by a polynomial $f$ of degree at most $q-1$ and $v\in\mathbb{F}_q$.

Projective deep-hole conjecture. The set

$$\{(u_f,v)\mid \deg(f)=k,\ v\in\mathbb{F}_q\}$$

is precisely the set of all deep holes of $PRS(q+1,k)$.

The conjecture is stronger than the covering-radius conjecture and aims to classify every deep hole of these projective Reed–Solomon codes. The paper proves the assertion under additional hypotheses and gives partial exclusion results for words with $\deg(f)\ge k+1$.