Some minimal cyclic codes over finite fields

In this paper, the explicit expressions for the generating idempotents, check polynomials and the parameters of all minimal cyclic codes of length t p n over F q are obtained, where p is an odd prime different from the characteristic of F q , t and n are positive integers with t ∣ ( q − 1 ) , gcd ( t , p ) = 1 and ord t p n ( q ) = ϕ ( p n ) . Our results generalize the main results in Pruthi and Arora (1997) and Arora and Pruthi (1999), which considered the cases t = 1 and t = 2 respectively. We propose an approach different from those in Pruthi and Arora (1997) and Arora and Pruthi (1999) to obtain the generating idempotents.