The MDS conjecture

Let $\mathbb{F}_q$ be a finite field, and let $[n,k]$ denote a linear code of length $n$ and dimension $k$ over $\mathbb{F}_q$ . If $1<k<q$ , then

n\le q+1,

except when $q$ is even and $k=3$ or $k=q-1$ , in which cases

n\le q+2.

The conjecture gives the expected maximum length of nontrivial maximum-distance-separable codes over finite fields. The paper uses known results in substantial parameter ranges, while the general assertion is presented as the MDS conjecture.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The MDS conjecture
Let $\mathbb{F}_q$ be a finite field, and let $\mathcal{C}\subseteq\mathbb{F}_q^n$ be a non-trivial MDS code of dimension $k$ . MDS conjecture. If such a code exists, then
$n\leq q+1,$
except when $q$ is even and $k=3$ or $k=q-1$ , in which case
$n\leq q+2.$
This conjecture gives a bound on the length of non-trivial MDS codes and, under its assumption, the preceding characterization covers almost all cases; the remaining cases can be handled individually.
source: Andreas Pyka and Violetta Weger, “On Optimal Homogeneous-Metric Codes”, arXiv:2603.27334 (2026).

Sources & referencesView supporting material

Primary source

Jun Zhang and Daqing Wan, “On Deep Holes of Projective Reed-Solomon Codes”, arXiv:1605.02423 (2016).

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