Classification of all MDS codes with a 2-transitive permutation group

Determine all maximum distance separable codes whose permutation automorphism group acts 2-transitively on the coordinate positions, including the affine-type cases not classified by the paper.

Background

The paper proves that an MDS code with a 2-transitive permutation automorphism group satisfies n\le q+1 in the stated parameter range and classifies the relevant almost-simple cases, showing that the surviving exceptional example is the scalar extension of the hexacode. Known affine-type examples include Reed–Solomon codes and additional codes arising from translation hyperovals and Segre arcs.

A complete classification is not obtained because the modular representation information needed for the affine-type groups is not generally available. The unresolved task is therefore to identify every MDS code whose coordinate permutation automorphism group is 2-transitive, rather than merely proving the length bound.

References

It is conceivably more challenging to determine all the MDS codes with a $2$-transitive permutation group $G$ of affine type, since such information is not available in general. We leave it an open problem to determine all the MDS codes with a $2$-transitive permutation group.

— On the lengths of MDS codes with a two-transitive permutation automorphism group  (2609.05292 - Deng et al., 4 Sep 2026) in Section 5, Conclusions