Cyclic codes of odd length over ℤ4 [u] / 〈u k 〉

Abstract

Let R=ź4u/źukź=ź4+uź4+ź+ukź1ź4$R=\mathbb{Z}_{4}[u]/ \langle u^k \rangle=\mathbb{Z}_{4}+u \mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$ (uk=0$u^{k}=0$), where k ź 2 is an positive integer. For any odd positive integer n, it is known that cyclic codes of length n over R are identified with ideals of the ring Rx/źxnź1ź$R[x]/\langle x^{n}-1\rangle$. In this paper, an explicit representation for each cyclic code over R of length n 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 of length n over R is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes of length n over R are investigated. As an application, some good quasi-cyclic codes of length 7k and index k over ź4 are obtained from cyclic codes over R = ź4 [u] / źukź when k = 2, 3, 4.