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 Fq\mathbb{F}_q is a fundamental question. The MDS conjecture gives the expected upper bound for non-trivial MDS codes, with possible q+2q+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.

References

One of the main problems in this area is the MDS conjecture, see the works of Ball and De~Beule , the book of Dougherty p.~284, and references there in. In particular, the conjecture says that an MDS linear over a prime field has length at most $n\leq q+1$ p.~284.

— Systems of parameters consisting of linear forms for monomial ideal quotients  (2609.30081 - Contreras et al., 24 Sep 2026) in Section 2, discussion preceding Theorem mds-minors

The celebrated MDS conjecture asserts, broadly speaking, that a non-trivial MDS code over $\mathbb{F}_q$ has length at most $q+1$, apart from certain exceptional parameter cases in which length $q+2$ may occur .

— New Constructions of Additive MDS TRS Codes  (2608.18904 - Bhagat et al., 19 Aug 2026) in Section 1, subsection “Background”

The MDS conjecture, which we state below, is a fundamental and long-standing problem in coding theory. Let $C$ be a nontrivial $[n,k]_q$ MDS code. Then $n\le q+1$, except when $q$ is even and $k\in{3,q-1}$, in which case $n\le q+2$.

— On the lengths of MDS codes with a two-transitive permutation automorphism group  (2609.05292 - Deng et al., 4 Sep 2026) in Introduction, Conjecture 1