Some cyclic codes of length \(2p^n\) (Q634132)

The authors consider the ideals of the ring \(\text{GF}(q)[x]/(x^{m}-1)\), where \(m=2p^n\), \(p,q\) are distinct primes, \(n\geq 1\) is an integer, and \(q\) has multiplicative order \(\varphi(m)/2 \pmod m\). The generator polynomials, the dimension, the minimum distance of cyclic codes of length \(2p^n\) generated by primitive idempotents of this ring are discussed.