The Structure Of A Class Of Constacyclic Codes Over Finite Chain Rings

Over the last decade, some researchers generalized linear codes over an arbitrary q-ary finite field F_q to finite rings. Wblfmann studied linear negacyclic and cyclic codes over Z₄ ,and showed that the Gray map image of a linear negacyclic code over Z₄ of length n is distance-invariant.Ling and Blackford introduced the Gray map over z_p^k+1 , and generalized most of the resultsin [19], [17]and [20]to the ring z_p^k+1 . In [9], Greferath and Schmidt generalized the Gray mapto finite chain rings.In this thesis, we shall continue the study codes over finite chain rings. We generalize some results in [12] to finite chain rings and obtain the following results.In chapter 2, Hamming weight enumerator and the generating matrix of a quasi-constant linear [q^k, k +1, (q -1)q^k-1 ] codes are given. The properties of the Gray map over finite chain rings are obtained. It is proved that the generalized Gray map is an isometry from (Rⁿ,d_hom ) toIn chapter 3, by considering two special unites ( (1 +γ^k) and (1 -γ^k)) in the finite chain rings,we characterize the structure of linear cyclic codes, (1 +γ^k) -cyclic codes and (1 -γ^k) -cyclic codes over finite chain rings, whereγis a fixed generator of the unique maximal ideal of the finite chain ring R, and the nilpotency index of is k + 1 .