Cyclic codes over $\mathbb{Z}_4[u]/\langle u^k\rangle$ of odd length
نویسندگان
چکیده
Let R = Z4[u]/〈u〉 = Z4 + uZ4 + . . . + uZ4 (u = 0) where k ∈ Z satisfies k ≥ 2. For any odd positive integer n, it is known that cyclic codes over R of length n are identified with ideals of the ring R[x]/〈x − 1〉. 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 over R of length n is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes over R of length n are investigated. When k = 4, some optimal quasi-cyclic codes over Z4 of length 28 and index 4 are obtained from cyclic codes over R = Z4[u]/〈u〉.
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