Minimal cyclic codes of length pnq

We consider the ring Rpnq=GF(ℓ)[x]/(xpnq−1), where p, q, ℓ are distinct odd primes, ℓ is a primitive root both modulo pn and q. Explicit expressions for all the (d+1)n+2 primitive idempotents are obtained, d=gcd(ϕ(pn),ϕ(q)), p∤(q−1). The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pnq over GF(ℓ) are also discussed.