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

TL;DR

This paper characterizes the Euclidean duals of certain cyclic codes generated by cyclotomic polynomials over finite fields, explicitly determining their minimum distances and confirming a prior conjecture.

Contribution

It provides a complete description of the dual codes' structure and proves the minimum distances are equal to 2 raised to the number of prime factors of n.

Findings

Dual codes' minimum distances are 2^{ω(n)}.

Structure of dual codes is explicitly described.

Conjecture on minimum distances is confirmed.

Abstract

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 $2^{\omega(n)}$.