On self-duality and hulls of cyclic codes over $\frac{{\mathbb{F}}_{2^m}[u]}{⟨u^k⟩}$ with oddly even length

Abstract

Let ${\mathbb{F}}_{2^m}$ be a finite field of $2^m$ elements, 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 $k$ is an integer satisfying $k\ge 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over ${\mathbb{F}}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ and a Gray map from ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ onto ${\mathbb{F}}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ are listed.