Let $\mathbb{ F}_{ 2^m}$ be a finite field of $2^m$ elements, 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 $k$ is an integer satisfying $k\geq 2$. 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.
Categories: cs.IT math.IT