2024/10/15 by Li Zhu, Zhu, Li, Hongfeng Wu +2
Computer Science · #Cellular Automata and Applications #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT) #Quantum-Dot Cellular Automata
paper · pdf · doi:10.48550/arxiv.2410.12122
openalex publication_date 2024/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Xn-1 and of Φn(X) over \mathbbFq given in \citeGraner. 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.