Primitive idempotents of irreducible cyclic codes of length \(p^nq^m\) (Q2865675)

Using the standard notations, this paper considers the case when \(\eta=p^nq^m\), where \(p, q\) are distinct odd primes and NEWLINE\[NEWLINEo(l)_{p^n}=\dfrac{\varphi(p^n)}{2},\;o(l)_{q^m}=\varphi(q^m),\;\gcd\left(\dfrac{\varphi(p^n)}{2}, \varphi(q^m)\right)=1.NEWLINE\]NEWLINE The complete set of \(2mn+2n+m+1\) cyclotomic cosets modulo \(p^nq^m\) has been obtained. The corresponding \(2mn+2n+m+1\) primitive idempotents in \(R_n\) have also been derived. As an illustration, an explicit expression for complete set primitive idempotents of irreducible cyclic codes of length 1089 has been described.