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 tpntpn over FqFq are obtained, where pp is an odd prime different from the characteristic of FqFq, tt and nn are positive integers with t∣(q−1)t∣(q−1), gcd(t,p)=1gcd(t,p)=1 and ordtpn(q)=ϕ(pn). Our results generalize the main results in Pruthi and Arora (1997) and Arora and Pruthi (1999), which considered the cases t=1t=1 and t=2t=2 respectively. We propose an approach different from those in Pruthi and Arora (1997) and Arora and Pruthi (1999) to obtain the generating idempotents.