Cyclic codes over ℤ4 + uℤ4

An explicit representation and enumeration for negacyclic codes of length 2^kn over \mathbb{Z}_4+u\mathbb{Z}_4

“…In the following, we think of Z4[x] x n −1 + u Z4[x] x n −1 (u 2 = 0) and R[x]/ x n − 1 as the same under the isomorphism Ψ. Then as a direct corollary of Theorems 3.4 and 3.7 in [7], we obtain the following conclusion: [3] and [23], the authors claimed that the number of all cyclic codes of length 7 over Z 4 + uZ 4 is equal to 7 3 = 343. Therefore, the formula for cyclic codes given by [3] and [23] is wrong.…”

“…There were some literatures on this kind of cyclic codes. Please refer to [3], [23] and [29], for examples. In these papers, the following results were given: ( †) "There are 7 m cyclic codes of length n over R" (Corollary 11 in [3] and Corollary 4.1 in [23]).…”

“…In fact, the number 7 m of cyclic codes over R with length n is wrong (see [23] and Theorem 4 in [29]). Now, we provided a new way different from the methods used in [3], [23] and [29] to study cyclic codes of odd length n over R = Z 4 + uZ 4 (u 2 = 0). To do this, we adopt the previous notations in Sections 1 and 2, for any integer 1 ≤ j ≤ r by Equations (1) and (2):…”

“…Then we correct a mistake in the mass formula for the number of negacyclic codes of length 2 k over Z 4 + uZ 4 in [4]. In Section 5, we give an explicit representation for all distinct cyclic codes of odd length n over Z 4 + uZ 4 and correct some mistakes in [3] and [23]. Section 6 concludes the paper.…”

Duadic codes over <inline-formula><tex-math id="M1">$ \mathbb{Z}_4+u\mathbb{Z}_4 $</tex-math></inline-formula>

“…Yildiz and Karadeniz [20] have studied linear codes and self-dual codes over Z 4 + uZ 4 . Cyclic codes over Z 4 + uZ 4 have been studied in [2,3,11,21]. Luo and Parampalli [11] have studied the structure of self-dual cyclic codes over Z 4 +uZ 4 and obtained some good self-dual cyclic codes over Z 4 via the Gray map.…”