Cyclic codes of length 4 pn over GF(q), where q is prime power of the form 4k+1

The explicit expression for the 4(n+1) primitive idempotents in FG (the group algebra of the cyclic group G of order 4 pn, where p is an odd prime, n≥1) over the finite field F of prime power order q, where qn is of the form 4k+1 and is a primitive root modulo p are obtained. The minimum distances, dimensions and the generating polynomials of the minimal cyclic codes generated by these primitive idempotents are also obtained.