The MDS conjecture

At least 9 years old · documented by 2016

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 1Other wordings

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).

References

Primary source

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

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

The general conjecture remains open, but an unverified submission claims a substantial new range of dimensions in odd characteristic.

Segre posed the conjecture in 1955. It predicts the maximum length of nontrivial MDS codes over finite fields, including the exceptional even-field cases.

Known results

Ball proved the conjecture for prime fields.
Ball and De Beule proved it for $q$ a power of the characteristic $p$ when $k<2p-2$ .
The conjecture is known when $k+1\le p$ and in further ranges near the dual boundary.
For even $q$ , the exceptional codes with parameters $[q+2,3]$ and $[q+2,q-1]$ exist; the boundary cases are settled.

September 28, 2026 claimed odd-characteristic progress

A submitted claim asserts that, for odd characteristic $p$ , the conjecture holds through $B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor$ , and by duality for $q+2-B(p,q)\le k\le q$ . It further claims these ranges cover asymptotically $1-\frac{1}{2p-3}$ of dimensions for fixed $p$ . This claim is unverified.

Current status (as of September 2026): Classical ranges and exceptional boundary codes are settled, while the general conjecture remains open and the new odd-characteristic range is only an unverified claim.

Sources

Partial progressA partial result proves the MDS conjecture over

\mathbb F_q

of odd characteristic for a range of dimensions linear in

q

. For fixed characteristic

p

, the proved ranges cover an asymptotic proportion

1-\frac{1}{2p-3}

of all dimensions.See full solutionHide full solution

A partial result is obtained for the MDS conjecture over finite fields of odd characteristic.

Let $q$ be a power of an odd prime $p$ , and set $B(p,q)= \left\lfloor \frac{(p-2)q+6p-10}{2p-3} \right\rfloor$ . The conjecture is proved for $2\le k\le B(p,q)$ and, by duality, for $q+2-B(p,q)\le k\le q.$ For fixed $p$ , these ranges are linear in $q$ and cover an asymptotic proportion $1-\frac{1}{2p-3}$ of all dimensions.

Preprint: https://doi.org/10.5281/zenodo.22908067

A linear-in- $q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic.