Let $${\mathbb {F}}_{2^m}$$ F 2 m be a finite field of characteristic 2 and $$R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}$$ R = F 2 m [ u ] / ⟨ u k ⟩ = F 2 m + u F 2 m + … + u k - 1 F 2 m ( $$u^k=0$$ u k = 0 ) where $$k\in {\mathbb {Z}}^{+}$$ k ∈ Z + satisfies $$k\ge 2$$ k ≥ 2 . For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring $$R[x]/\langle x^{2n}-1\rangle $$ R [ x ] / ⟨ x 2 n - 1 ⟩ . In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.