MDS-extension conjecture for Reed–Solomon codes

Let $q$ be a prime power and let $2\leq m\leq q-2$. An MDS extension by one digit of an $m$-dimensional Reed–Solomon code with evaluation set $GF(q)$ is an MDS code obtained by adding one coordinate; a generalized Reed–Solomon (GRS) code is the corresponding generalized evaluation code. Equivalently, a $(q+1)$-arc in $PG(m-1,q)$ has $q$ points on a normal rational curve (RNC). MDS-extension conjecture. Except when $m=3$ and $q$ is even, every such MDS extension is GRS; equivalently, every $(q+1)$-arc in $PG(m-1,q)$ with $q$ points on an RNC has all its points on the RNC. This is the geometric restatement of the Cheng–Murray conjecture. The paper establishes the claim in the range corresponding to $k\geq\lfloor(q-1)/2\rfloor$ and relates the remaining cases to questions about arcs and MDS codes.