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

Abstract

Let ${\mathbb{F}}_{2^m}$ be a finite field of characteristic $2$ and $R={\mathbb{F}}_{2^m}[u]/⟨u^k⟩={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+\dots +u^{k-1}{\mathbb{F}}_{2^m}$ ($u^k=0$) 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 $R[x]/⟨x^{2n}-1⟩$. 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. (AAECC-1522)

arXiv commentAAECC-1522