On the Euclidean duals of the cyclic codes generated by cyclotomic polynomials
Anuj Kumar Bhagat, Ritumoni Sarma
TL;DR
This work determines the Euclidean minimum distance of the dual of the cyclic code $\mathcal{C}_n$ generated by the cyclotomic polynomial $Q_n(x)$ over $\mathbb{F}_q$ for all $n$ with $\gcd(n,q)=1$. By leveraging the order of polynomials, cyclotomic cosets, and a multiplicative framework via direct products and the CRT, the authors prove that $d(\mathcal{C}_n^{\perp})=2^{\omega(n)}$, completing a conjecture posed in prior work. The proof proceeds first through the prime-power case, where $d(\mathcal{C}_{p^a}^{\perp})=2$, and then extends to general $n$ by showing that the dual distance factors multiplicatively according to the distinct prime factors of $n$. They also establish a decomposition of $\mathcal{C}_n^{\perp}$ and discuss the dual of the related code $\mathcal{C}_{n,1}$, noting that the corresponding distance conjecture remains open for composite $n$. The results provide a precise link between dual distance and the prime-factor structure of $n$, with implications for code design and related algebraic coding theory questions.
Abstract
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}.