Euclidean-dual distance conjecture for the cyclic code generated by $Q_n(x)Q_1(x)$

Let $n$ be a composite positive integer, let $q$ be a prime power with $\operatorname{char}(\mathbb{F}_q)\nmid n$ , and let $\mathcal{C}_{n,1}$ be the cyclic code over $\mathbb{F}_q$ generated by $Q_n(x)Q_1(x)$ , where $Q_n(x)$ is the $n$ th cyclotomic polynomial. Write $\omega(n)$ for the number of distinct prime divisors of $n$ . Euclidean-dual distance conjecture. The Euclidean dual $\mathcal{C}_{n,1}^{\perp}$ has parameters

[n,\varphi(n)+1,2^{\omega(n)}].

Equivalently, its minimum distance is $2^{\omega(n)}$ . The paper states this as remaining conjectural; the analogous assertion for the dual of the code generated by $Q_n(x)$ is proved, while the claim for $\mathcal{C}_{n,1}^{\perp}$ remains open.

Sources & referencesView supporting material

Primary source

Anuj Kumar Bhagat and Ritumoni Sarma, “On the Euclidean duals of the cyclic codes generated via cyclotomic polynomials”, arXiv:2601.03165 (2026).

1 week ago

Progress summary

Refreshed 4 days ago

Claimed solved

A January 2026 preprint claims to settle the conjecture, but the available evidence is internally inconsistent and no independent verification was found.

The conjecture predicts that the Euclidean dual of the cyclic code generated by $Q_n(x)Q_1(x)$ has minimum distance $2^{\omega(n)}$ and parameters $[n,\varphi(n)+1,2^{\omega(n)}]$ . The original proposer and date are not identified in the retrieved sources.

Known results

The analogous distance formula for the Euclidean dual of the code generated by $Q_n(x)$ is proved in the January 2026 preprint.
The $\mathcal{C}_{n,1}^{\perp}$ assertion was previously proved only when $n$ is a product of two distinct prime powers.

January 2026 claimed resolution

The arXiv record and a mirror state that the preprint proves the target distance formula for both relevant dual codes, which would resolve the conjecture. However, the supplied PDF search result describes the $\mathcal{C}_{n,1}^{\perp}$ case as still conjectural; no independent proof, correction, or referee assessment was found.

Current status (as of August 2026): A January 2026 preprint claims the conjecture is proved, but the claim is unverified because the retrieved record is internally inconsistent and lacks independent confirmation.

Sources

No solutions have been posted yet.