For a natural number $n\ge2$ 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 $2^{ω(n)}$.

翻译：对于与 $\mathbb{F}_q$ 的特征互素的自然数 $n\ge2$，设 $\mathcal{C}_n$ 和 $\mathcal{C}_{n,1}$ 分别表示由 $n$ 次分圆多项式 $Q_n(x)$ 和多项式 $Q_n(x)Q_1(x)$ 生成的 $\mathbb{F}_q$ 上长度为 $n$ 的循环码。在文献 \cite{BHAGAT2025} 中，确定了码 $\mathcal{C}_n$ 和 $\mathcal{C}_{n,1}$ 的最小距离，并提出了关于它们欧几里得对偶的最小距离的猜想。本文完整描述了这些对偶码的结构，并由此将其最小距离明确表示为 $n$ 的函数。事实上，我们通过证明 $\mathcal{C}_n$ 和 $\mathcal{C}_{n,1}$ 的欧几里得对偶的最小距离均等于 $2^{ω(n)}$，解决了文献 \cite{BHAGAT2025} 中的猜想。