Research On Asymptotic Property And Weight Distribution Of Linear Codes Over Finite Rings

In this thesis,we mainly study the construction of three-weight codes over finite rings,the asymptotic properties of quasi-cyclic codes and quasi-twisted codes over finite fields,and the maximum number L(k,q)of non-zero weights a linear codes of dimension k can have.The details are as follows:1.The trace codes over finite non-chain ring R1 are studied.Using char-acter sums,we give the weight distribution of the trace codes.Through proper Gray mapping,we obtain two classes of three-weight codes over finite fields.When m is even,the Gray image of Cm is a class of linear codes with three non-zero weights over finite fields.When m is odd,the Gray image of Cm is also a class of linear codes with three non-zero weights over finite fields,and reaches the Griesmer bound,i.e.optimal.Furthermore,we discuss the minimal distance of dual codes of Cm.Combining the weight distribution of two types of three-weight codes,we verify that the codewords of Gray image constructed are minimal,and it can be applied in secret sharing schemes.2.Some special classes of quasi-cyclic codes and quasi-twisted codes over finite fields are studied,which are double(negacirculant)circulant codes and four(negacirculant)circulant codes.We study the asymptotic behavior of self-duality,LCD double(negacirculant)circulant codes and four(negacirculant)circulant codes as follows:(i)Counting formulas of self-dual double(negacirculant)circulant codes and four(negacirculant)circulant codes.(ii)Counting formulas of LCD double(negacirculant)circulant codes and four(negacirculant)circulant codes.When we control the decomposition of xn±1 into some special decomposition,these family codes are proved to be asymptotically good.3.We study the maximum number of non-zero weights of linear codes of dimension k over Fq.We give the exact expressions of L(2,q)and L(k,2).When both k and q are greater than 2,we give the upper and lower bounds of L(k,q).Furthermore,we study the upper and lower bounds of non-zero weights of a linear codes of dimension k with fixed length n over Fq.Then,we discuss the asymptotic behavior of L(k,q)and L(n,k,q).Finally,the maxi-mum number of extremal function with non-zero distances of non-linear codes and non-linear codes with fixed length n over alphabet Aq,denoted by N(M,q)and N(n,M,q),respectively,are researched.