On The Structures And Applications Of Several Types Of Algebraic Codes Over Finite Fields And Finite Rings

In 1994,it was found that some important non-linear binary codes can be con-structed by linear codes with special structure under Gray maps.Afterwards,coding scholars began to study the theory of error-correcting codes over finite rings.Based on the existing research results,this thesis provides a further research on the theory of error-correcting codes over finite rings and finite fields including the covering radius of linear codes over a finite ring,the algebraic structures of skew constacyclic codes and skew cyclic codes over finite rings and the construction of optimal codes over finite fields.Finally,some linear codes with better parameters are obtained.As an applica-tion,we construct some new quantum error-correcting codes.The details are given as follows:In Chapter 1,we introduce the history and research status of error-correcting codes and briefly summarize the main work done in this thesis.In Chapter 2,let R1=F2+vF2={0,1,v,1+v},where v2=v.In this section,we determine the bounds on the covering radius of repetition codes,simplex codes and Mac Donald codes over F2×R1under the Chinese Euclidean distance.In Chapter 3,let q be a prime power with gcd（q,6）=1.Let R2=Fq2+uFq2+vFq2+uvFq2,where u2=u,v2=v and uv=vu.In this section,we study linear skew constacyclic codes over Fq2×R2.We discuss the structural properties and determine the generator polynomials and the minimal generating sets of this family of codes.More-over,by establishing a Gray map preserving the Hermitian orthogonality from Fq2α×R2β to Fq2α+4β,we obtain Hermitian dual-containing codes over Fq2as Gray images of linear skew constacyclic codes over Fq2×R2,whereαandβare positive integers.Finally,as an application,we get some new quantum error-correcting codes with better parameters by Hermitian construction.In Chapter 4,let R3=Fq+uFq,where u2=0.In this section,we study linear skew cyclic codes over Fq×R3.Then,we determine the generator polynomials and the minimal generating sets of these codes.Further,the dual codes of separable linear skew cyclic codes are also presented.Finally,by establishing a Gray map from Fq×R3 to Fq3,we obtain some optimal linear codes over Fq.In Chapter 5,we study the construction of Fq- linear codes over Fql,where l is a prime number.First,through a special kind of Vandermonde matrices,called Fourier matrices,the maximum distance separable（MDS）Fq- linear codes over Fq2are giv-en.Then,by a canonical decomposition of constacyclic Fq- linear codes over Fql,we construct MDS Fq- linear codes over Fql.In Chapter 6,we summary the content of this thesis and propose several issues for further study.