Wu–Hong revised conjecture on deep holes of primitive Reed–Solomon codes

Let $\mathbb{F}_q$ be a finite field, let $D=\mathbb{F}_q^*$ be the evaluation set of a primitive Reed–Solomon code with code length $n=q-1$, dimension $k$, and codebook $\mathcal{C}$. Let $u$ be a received word represented by its interpolating polynomial $u(x)$. A word is a deep hole when $d(u,\mathcal{C})=n-k$. Wu–Hong revised conjecture. All deep holes for primitive Reed–Solomon codes are those words $u$ represented by

$$u(x)=a x^k+v(x) \quad\text{or}\quad u(x)=a x^{q-2}+v(x),$$

where $a\ne 0$ and $\deg(v(x))\leq k-1$. The second family is known to consist of deep holes, and the conjecture revises the Cheng–Murray claim in light of that family; the full classification remains open in the stated generality.