RNC completeness conjecture for MDS extensions

Let $q$ be a prime power and let $m$ be an integer for which the stated MDS codes are defined. An $[q+1,m]_q$ Reed–Solomon code corresponds to a normal rational curve (RNC) in $PG(m-1,q)$, and an MDS extension by one coordinate is an $[q+2,m]_q$ MDS code extending it. RNC completeness conjecture. There is no $[q+2,m]_q$ MDS code extending an $[q+1,m]_q$ Reed–Solomon code, except when $q$ is even and $m=3$ or $m=q-1$; equivalently, the RNC in $PG(m-1,q)$ is a complete arc except in those cases. The source notes that this conjecture is implied by the MDS conjecture and proves that the preceding MDS-extension conjecture implies it.