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 $n\ge 2$. Some lower bounds on the minimum distances of irreducible cyclic codes of length $2p^n$ over $GF(q)$ are also obtained.