Palestine Journal of Mathematics Vol. 4 (Spec. 1) (2015) , 540–546 © Palestine Polytechnic University-PPU 2015 On Skew Cyclic and Quasi-cyclic Codes Over F 2 + uF 2 + u 2 F 2 Abdullah Dertli, Yasemin Cengellenmis and Senol Eren Communicated by Ayman Badawi MSC 2010 Classifications: Primary 94B05; Secondary 94B15. Keywords and phrases: Cyclic codes,quasi-cyclic codes, skew cyclclic codes, skew quasi-cyclic codes, Finite rings. Abstract We construct a new Gray map from S n to F 3n 2 where S = F 2 + uF 2 + u 2 F 2 ,u 3 = 1. It is both an isometry and a weight preserving map. It was shown that the Gray image of cyclic code over S is quasi-cyclic codes of index 3 and the Gray image of quasi-cyclic code over S is l-quasi-cyclic code of index 3. Moreover, the skew cyclic and skew quasi-cyclic codes over S introduced and the Gray images of them are determined. 1 Introduction By using generator polynomials in skew polynomial rings, there are a lot of works about gen- eralizing notions of cyclic, constacyclic, quasi-cyclic codes to skew cyclic, skew constacyclic, skew quasi-cyclic codes respectively. Skew polynomial rings form an important family of non-commutative rings. There are many applications in the construction of algebraic codes. As polynomials in skew polynomial ring exhibit many factorizations, there are many more ideals in a skew polynomial ring than in the commutative case. So the research on codes have result in the discovery of many new codes with better Hamming distance. Works began with D. Boucher, W. Gieselmann, F. Ulmer’s paper in [1]. They generalized the notion of cyclic codes. They gave many examples of codes which improve the previously best known linear codes. In [3], D. Boucher and F. Ulmer studied linear codes over finite fields obtained from left ideals in a quotient ring of a (non-commutative) skew polynomial ring. They show how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is generalization of the θ−cyclic codes discussed in [1]. Moreover, they shown that the dual of a θ−cyclic code is still θ−cyclic. D. Boucher, P. Sole and F. Ulmer generalized the construction of linear codes via skew poly- nomial rings by using Galois rings instead of finite fields as coefficients. Codes that are principal ideals in quotient rings of skew polynomial rings by a two side ideals were studied in [2]. In [4], they studied a special type of linear codes called skew cyclic codes in the most general case. They shown that these codes are equivalent to either cyclic or quasi-cyclic codes. Skew polynomial rings over finite fields and over Galois rings had been used to study codes. In [9], they extended this concept to finite chain rings. The structure of all skew constacyclic codes is completely determined. In [11], T. Abualrub, P. Seneviratre studied skew cyclic codes over F 2 + vF 2 ,v 2 = v. In [10], T. Abualrub, A. Ghrayeb, N. Aydın, ˙ I. ¸ Siap introduced skew quasi-cyclic codes. They obtained several new codes with Hamming distance exceeding the distance of the previously best known linear codes with comparable parameters. In [7], M. Bhaintwal studied skew quasi-cyclic codes over Galois rings. In [8], they investigated the structures of skew cyclic and skew quasi-cyclic of arbitrary length over Galois rings. They shown that the skew cyclic codes are equivalent to either cyclic and quasi-cyclic codes over Galois rings. Moreover, they gave a necessary and sufficient condition for skew cyclic codes over Galois rings to be free. Jian Gao, L. Shen, F. W. Fu studied a class of generalized quasi–cyclic codes called skew generalized quasi-cyclic codes. They gave the Chinese Remainder Theorem over the skew poly- nomial ring which lead to a canonical decomposition of skew generalized quasi-cyclic codes. Moreover, they focused on 1-generator skew generalized quasi-cyclic code in [6]. J.F. Qian et. al. introduced linear (1 + u)-constacyclic codes and cyclic codes over F 2 + uF 2 and characterized codes over F 2 which are the Gray images of (1 + u)-constacyclic codes or cyclic codes over F 2 + uF 2 in [5 ]. It was introduced (1 − u m )-constacyclic codes over F 2 + uF 2 + ... + u m F 2 and characterized codes over F 2 in [12 ]. This paper is organized as follows. In section 2, we give some basic knowledges about the