Skew Cyclic Codes Over Finite Ring And Its Application

On the basis of the general polynomial ring,the ring become skew polynomial ring after attemping automorphism mapping to it.The introduce of automorphism mapping changes the multiplication of elements in the ring,this skew polynomial ring become non commutative.The code words of skew polynomial ring have a larger space for searching.As a new generalization of cyclic codes,skew cyclic codes have attracted the attention of many scholars at home and abroad.And a new branch of coding theory in finite fields and finite rings has formed.According to the research results of Abualrub,Ashraf,Boucher,Zhu Shixin and other scholars,with the help of finite rings,polynomial ideal,module etc.algebra theory and cyclic code theory,mainly discuss structure and properties of skew cyclic codes over ring R = F_q +uF_q+…u^k-1F_q（q is the prime power）.The following aspects are studied in detail:In this paper,we investigate the structure,properties and applications of cyclic codes over the finite ring R by constructing a new automorphism mapping and Gray mapping on the ring.We discuss it in three cases.For the 1st case of ring R,a new automorphism mapping is introduced to R,then gives the center of the non commutative ring R,and the structure and properties of the skew cyclic codes of arbitrary length over R are researched.At the same time,gives the definition of Gray mapping φ from R to F_q,and discuss on the properties of skew cyclic codes underφ.For the 2nd case of the ring R（p is odd prime,uk = 0）,according to the characteristics of elements in this ring,introduce a new automorphism mapping to the ring R,combining with the previous Gray mapping,we discuss the structure and properties of skew cyclic codes over ring R.For the 3rd case,we research the structure,properties and direct sum decomposition of skew cyclic codes over ring F5 + uF5 +…+u^k-1F5（u3＝4u）.In addition,we study the relationship of skew cyclic codes,cyclic codes and quasi-cyclic codes over the ring,and gives skew cyclic codes of different length.