Irreducible Cyclic Codes of Length \(2p^n\)

Abstract

Let \(q\) be an odd prime power and \(p\) be an odd prime with \(gcd(p, g) = 1\). Let the order of \(g\) modulo \(p\) be \(f\) and \(gcd(\frac{p-1}{f}, q) = 1\). Here explicit expressions for all the primitive idempotents in the ring \(R_{2p^n} = GF(q)[x]/(x^{2p^n} – 1)\), for any positive integer \(n\), are obtained in terms of cyclotomic numbers, provided \(p\) does not divide \(\frac{q^f-1}{2p}\), if \(n \geq 2\). Some lower bounds on the minimum distances of irreducible cyclic codes of length \(2p^n\) over \(GF(q)\) are also obtained.