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

Abstract

Let $$\mathbb {F}_{2^m}$$ be a finite field of $$2^m$$ elements and denote $$R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle $$ $$=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}$$ ( $$u^k=0$$ ), 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. In particular, a generator matrix is provided for the self-dual 2-quasi-cyclic code of length 4n over $$\mathbb {F}_{2^m}$$ derived by an arbitrary 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.