On the idempotents of cyclic codes over F2t

Abstract

We study cyclic codes of length $n$ over ${\mathbb{F}}_{2^t}$. Cyclic codes can be viewed as ideals in ${\mathcal{R}}_n={\mathbb{F}}_{2^t}[x]/(x^n-1)$. It is known that there is a unique generating idempotent for each ideal. Let $e(x)\in {\mathcal{R}}_n$. If $t=1$ or $t=2$, then there is a necessary and sufficient condition that $e(x)$ is an idempotent. But there is no known similar result for $t\ge 3$. In this paper we give an answer for this problem.