On the Euclidean duals of the cyclic codes generated by cyclotomic polynomials

In this article, we determine the minimum distance of the Euclidean dual of the cyclic code $\mathcal{C}_n$ generated by the $n$th cyclotomic polynomial $Q_n(x)$ over $\mathbb{F}_q$, for every positive integer $n$ co-prime to $q$. In particular, we prove that the minimum distance of $\mathcal{C}_{n}^{\perp}$ is a function of $n$, namely $2^{ω(n)}$. This was precisely the conjecture posed by us in \cite{BHAGAT2025}.