Research On Constacyclic Codes Over F_q+uF_q+…+u^k-1F_q

Coding theory over finite fields is not only more mature,and has been widely used in practice.In1994,coding theory over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes under the Gray map by Hammons et al.In particular,codes over finite rings have been given more attention.In this paper,we investigate the structure and properties of a class constacyclic codes of an arbitrary length over Fq+uFq+…+uk-1Fq.The details are given as follows:1. we study (1+uβ)-constacyclic codes over the ring R=Fq+uFq+…+uk-1Fq of an arbritary length.We determine the number of distinct (1+uβ)-constacyclic codes over R of length N=psn,and the number of codewords in each such code.We derive the structure of (1+uβ)-constacyclic dual codes over the ring R.2. A class of (1+λu)-constacyclic MDS codes over the ring Fq+uFQ+…+uk-1Fq by using RS codes over Fpm of length n=pm-1was obtained。By extended RS codes over Fpm,we obtained some classes of (1+λu)-constacyclic MDS codes over this ring of length n=pm+1.When k=2,some classes of cyclic MDS codes over Fpm+uFpm of length n=pm-1and n=pm+1were obtained.3. In this paper,we study all constacyclic codes over the ring Fpm+uFp of arbitrary lengths.Using the Res(C) and Tor(C),we obtain a unique set of generators of (α+β)-constacyclic codes and λ-constacyclic codes. we derive the structure of these constacyclic dual codes.