MDS conjecture for general codes

Let $G=([n],E)$ be a $k$-uniform hypergraph, and let $q(G)$ be the smallest alphabet size for which there are encoding and decoding functions recovering every message from the symbols on every edge of $G$. For the complete $n$-vertex $k$-uniform hypergraph $\kappa_{n,k}$, $q(\kappa_{n,k})$ is the minimum alphabet size of a general $(n,k)$ MDS code. For integers $k<q\neq 6$, let $n(q,k)$ be the largest integer $n$ such that $q(\kappa_{n,k})\leq q$. MDS conjecture for general codes.

$$n(q,k)\leq\begin{cases}q+2 & \text{if }4\mid q\text{ and }k\in\{3,q-1\},\q+1 & \text{otherwise.}\end{cases}$$

This is the classical upper bound on the length of general MDS codes over an alphabet of size $q$. The supplied status evidence says that the linear version over prime fields has been proved, but it does not establish resolution of this general-code statement; its database status is therefore left open.