Research On Some Problems Of Constacyclic Codes Over The Finite Chain Ring

In this pape r,we mainly research on the structure and properties of some class of constacyclic codes and their duals over finite chain ringR=Fpm+uFpm+…+ukFpm.In chapter1,we describe the backgroud and the research progress of constacyclic codes over R=Fpm+uFpm+…+ukFpm.And then we illuStrate the main content ofthis paper.In chapter2,we introduce some definitions and properties about groups,rings and polynomials.And based on these,we introduce some basic definitions and theorems on the linear codes,cyclic codes and constacyclic codes,which will be used to study the constscyclic codes overR=Fpm+uFpm+…+ukFpm.In chapter3,we give some necessary notations first.Then we give the discrete Fourier transformation about theδ一contacyclic codes of lengthN.We also give the specific formula of discrete Fourier transforrmation with some special parameters.And then we discuss the struIcture and properties of the δ—contacyclic codes of lengthN in detail.Secondly,we discuss the structure and properties of the (1+αu)―contacyclic codes of length N over R=Fpm+ufpm+…+ukFpm, specifically.Generator polynomials and enumeration of all the different (1+αu)―contacyclic codes of length Nare obtained.And we prove that the dual of (1+αu)―contacyclic code of length N is a(1+αu)-1―constacyclic code.And then,we also discuss the generator polynomials and enumeration of all the different(1+αu)-1―contacyclic codes of lengthN.Necessary and suffcient conditons for(1+αu)一constacyclic codes to be self-dual are also obtaine d.At last,we study (β+γu)一constacyclic codes of lengthNover R=Fpm+uFpm+…+ukFpm.We build a one-to―one and weight preserving correspondence between (1+a")一contacyclic codes and(β+γu)一constacyclic codes of lengthNover R=Fpm.+uFpm+…+ukFpm.By use ofthis correspondence,we also similarly obtain structure and properties of dual codes and self-dual codes0f(β+γu)一constacyclic codes of lengthNover R=Fpm+uFpm+…+ukFpmIn chapter4,we give a conclusion of this paper,and then,we describe the further research about the (β+γu)一constacyclic codes of length N over R=Fpm+uFpm+…+ukFpm.