Explicit representatives and sizes of cyclotomic cosets and their application to cyclic codes over finite fields

Cyclotomic coset is a classical notion in the theory of finite fields 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 Φ n ( X ) over F q given in [4] . 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.