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 F 2 m be a finite field of 2 m elements, andFor 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 selfdual and 2-quasi-cyclic code of length 4n over F 2 m derived by every self-dual cyclic code of length 2n over F 2 m + uF 2 m and a Gray map from F 2 m + uF 2 m onto F 2 2 m . Finally, the hull of each cyclic code with length 2n over F 2 m + uF 2 m is determined and all distinct self-orthogonal cyclic codes of length 2n over F 2 m + uF 2 m are listed.On self-duality and hulls of cyclic codes overu k with oddly even length 3On self-duality and hulls of cyclic codes overwith oddly even length 5On self-duality and hulls of cyclic codes over F 2 m [u] u k with oddly even length 9