Cyclic codes over ${\mathbb {F}}_{2^m}[u]/\langle u^k\rangle$ of oddly even length

Abstract

Let ${\mathbb {F}}_{2^m}$ be a finite field of characteristic 2 and $R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}$ ($u^k=0$) where $k\in {\mathbb {Z}}^{+}$ satisfies $k\ge 2$. For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring $R[x]/\langle x^{2n}-1\rangle$. In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.