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}.

翻译：本文中，我们确定了由第$n$个分圆多项式$Q_n(x)$在有限域$\mathbb{F}_q$上生成的循环码$\mathcal{C}_n$的欧几里得对偶码的最小距离，其中正整数$n$与$q$互素。特别地，我们证明了$\mathcal{C}_{n}^{\perp}$的最小距离是$n$的函数，即$2^{ω(n)}$。这正是我们在文献\cite{BHAGAT2025}中提出的猜想。