Research On The MacWilliams Identity Of Linear Codes Over Z₄+vZ₄ And Quasi-twisted Codes Over F₂[u]/(u^k+1)

Weight enumerator is one of the important parameters of a code. The Mac Williams identity for a code is a remarkable result that describes how the weight distribution of a code and that of its dual are related to each other.Quasi-twisted codes over finite fields form an important class of block codes that includes cyclic codes, quasi-cyclic codes and constacyclic codes as special cases. Therefore quasi-twisted codes deserve a careful study.The article divides into two parts. In the first section, an identity for m-spotty weight distribution between the linear codes and its dual over Z₄+vZ₄ is studied. Firstly, spotty weight and m-spotty weight enumerator of byte error control codes over the ring Z₄+vZ₄ are introduced. Moreover, the Mac Williams identity for m-spotty weight enumerator of the m-spotty byte error control codes over Z₄+vZ₄ is proved. Also, an example is given to illustrate the main theorem. In the second part, （1+ uk）-quasi-twisted codes over the ring R= F₂ [u]/(u^k+1) are introduced. The key idea is to consider a （1+u^k）-quasi-twisted code over R as a linear code over Rm= R[x]/（xm—（1+u^k））. By using the Chinese remainder theorem or the discrete Fourier transform, the ring R[x]/（xm—（1+u^k）） can be decomposed into a direct product of finite chain rings. The inverse transform of the discrete Fourier transform induces a （1+u^k）-quasi-twisted code construction from codes of lower lengths.