Study On Some Problems Of F_qF_q[u,v]-additive Codes

In recent years,the additive codes over finite rings have attracted much more attentions by researchers,because they have good algebraic structures.In this paper we mainly focus on the study of some problems of additive codes over the finite ring FqF1[u,v],where Fq[u,v]=Fq+uFq+vFq+uvFq,q is the power of a prime,u2=v2=0,uv=vu.Simultaneously,we construct some few weight codes over the ring Z3Z3[v].The details are given as follows:In Chapter 3 we first introduce the concepts of FqFq[u,v]-additive codes and Gray map,and then the Hamming weight enumerators,the Lee weight enumerators and the symmetrical weight enumerators of FqFq[u,v]-additive codes and their relationships are discussed.Moreover,when q is the power of 2,we obtain the MacWilliams identities between a FqFq[u,v]-additive code and its dual code on Lee weights.In Chapter 4 we discuss the m-spotty RT weights of FqFq[u,v]-additive codes.By defining a new inner product,the MacWilliams identities with respect to the RT weights between a FqFq[u,v]-additive code and its dual code are obtained,when q is the power of 2.An example is given to show the validity of the above identities.In Chapter 5 we investigate the FqFq[u,v]-additive cyclic codes.The algebraic structure of this type of additive cyclic codes is studied and the minimal generator sets of polynomials Rα,β[x]are attained.We show that the dual of the FqFq[u,v]-additive cyclic codes are also additive cyclic codes.Finally,in Chapter 6 we study one-Lee weight and two-Lee weight Z3Z3[v]-additive codes,where Z3[v]=Z3+vZ3,v2=v.Furthermore,an complete classification of one-Lee weight and two-Lee weight Z3Z3[v]-additive codes is given and the structure of one-Lee weight Z3Z3[v]-additive codes is obtained.