Minimal cyclic codes of length pnq

Explicit expressions for all the 3n+2 primitive idempotents in the ring Rpnq=GF(ℓ)[x]/(xpnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pn and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pnq over GF(ℓ) are also discussed.