Conjecture on the automorphism group of symmetric Reed–Muller codes

Let $J_n$ be the $n \times n$ all-one matrix, $I_n$ the $n \times n$ identity matrix, and $\mathcal{P}_n$ the set of permutation matrices of order $n$ . Define

M=\left\{P\bigl((b-a)I_n+aJ_n\bigr)\mid P\in\mathcal{P}_n,\a,b\in\mathbb{F}_q,\a\neq b,\a\neq(1-n)b\right\}\subset GL(n,q).

Automorphism-group conjecture. For $q>r>\frac{n(n-1)}{2}$ , the automorphism group of $SRM_q[n,r]$ contains a subgroup isomorphic to $M$ ; equivalently, $SRM_q[n,r]$ is invariant under the transformations in $M$ . The complete determination of $\operatorname{Aut}(SRM)$ for arbitrary $n>3$ remains open, since the paper proves the corresponding automorphism groups only for $n=2$ and $n=3$ .

Sources & referencesView supporting material

Primary source

Sibel Kurt Toplu, Talha Arikan, Pinar AydoğDu and OğUz Yayla, “On a Group Under Which Symmetric Reed-Muller Codes are Invariant”, arXiv:2401.11496 (2024).

1 week ago

Progress summary

Refreshed 4 days ago

Open

The proposed symmetry group is verified only in two small dimensions; the general claim remains open, with no public proof or counterexample found.

The conjecture asserts that, when $q>r>\frac{n(n-1)}{2}$ , the symmetric Reed–Muller code $SRM_q[n,r]$ is invariant under the stated group $M$ . The general case beyond the dimensions already analyzed remains unresolved.

Known results

For $n=2$ , the paper determines the relevant transformations and proves the proposed subgroup inclusion under its stated parameter conditions.
For $n=3$ , it determines the corresponding invariant transformation group.
For arbitrary $n>3$ , neither the conjectured inclusion nor completeness of the proposed group is established.

January 2024 paper

The paper “On a group under which symmetric Reed–Muller codes are invariant” formulates the general conjecture after proving the cases $n=2$ and $n=3$ . Later publication and repository records report no proof, counterexample, or verification resolving the general case.

Current status (as of August 2026): The cases $n=2$ and $n=3$ are treated, but the automorphism-group conjecture and complete determination of $\operatorname{Aut}(SRM)$ for $n>3$ remain open.

Sources

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