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

For a natural number $n\ge 2$ which is co-prime to Char$({\mathbb{F}}_q)$, let ${\mathcal{C}}_n$ and ${\mathcal{C}}_{n,1}$ denote the cyclic codes of length $n$ over ${\mathbb{F}}_q$ generated by the $n$-th cyclotomic polynomial $Q_n(x)$ and the polynomial $Q_n(x)Q_1(x)$, respectively. In \cite{BHAGAT2025}, the minimum distances of the codes ${\mathcal{C}}_n$ and ${\mathcal{C}}_{n,1}$ were determined, and a conjecture regarding the minimum distances of their Euclidean duals was proposed. In this article, we completely describe the structure of these dual codes and as a consequence, we find their minimum distances explicitly as functions of $n$. In fact, we resolve the conjecture in \cite{BHAGAT2025} by proving that the minimum distance of the Euclidean dual of each of ${\mathcal{C}}_n$ and ${\mathcal{C}}_{n,1}$ is equal to $2^{\omega (n)}$.