MDS conjecture on the maximum length of nontrivial MDS codes

Prove the MDS conjecture that every non-trivial MDS code over the finite field \(\mathbb{F}_q\) has length at most \(q+1\), except in certain exceptional parameter cases where length \(q+2\) may occur.

Background

The paper places additive MDS codes within the broader theory of MDS codes and notes that determining the largest possible length of an MDS code over $\mathbb{F}_q$ is a fundamental question. The MDS conjecture gives the expected upper bound for non-trivial MDS codes, with possible $q+2$-length exceptions for certain parameters. The paper further notes that substantial progress has been made, including a proof for prime fields, but does not claim that the conjecture is resolved in full generality.