# On self-duality and hulls of cyclic codes over F_2^m[u]/〈 u^k〉 with oddly even length

Let F_2^m be a finite field of 2^m elements, and R=F_2^m[u]/〈 u^k〉=F_2^m+uF_2^m+...+u^k-1F_2^m (u^k=0) where k is an integer satisfying k≥ 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 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^m^2. 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.

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