🤖 AI Summary
This paper systematically characterizes the structure of $q$-cyclotomic cosets modulo $n$ over the finite field $mathbb{F}_q$, thereby strengthening the theoretical foundation of cyclic codes. Methodologically, it establishes, for arbitrary $q$ and $n$, a unified explicit formula for canonical representatives of cyclotomic cosets and a closed-form expression for their exact sizes; introduces the novel *2-adic cyclotomic system* framework to uncover deep structural regularities; and constructs a direct mapping from cyclotomic information to key cyclic code parameters—including generator polynomials and self-duality properties. Main contributions include: an optimized Graner-type factorization formula yielding fine-grained factorizations of $X^n - 1$ and the $n$-th cyclotomic polynomial $Phi_n(X)$ over $mathbb{F}_q$; a complete classification of all generator polynomials of cyclic codes of length $n$ over $mathbb{F}_q$; and necessary and sufficient conditions for self-dual cyclic codes, along with their precise enumeration.
📝 Abstract
Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let $q$ be a prime power, and $n$ be a positive integer coprime to $q$. In this paper we determine explicitly the representatives and the sizes of all $q$-cyclotomic cosets modulo $n$ in the general settings. We introduce the definition of $2$-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the $2$-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of $q$-cyclotomic cosets modulo $n$, we improve the formulas for the factorizations of $X^{n}-1$ and of $Phi_{n}(X)$ over $mathbb{F}_{q}$ given in cite{Graner}. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.