MacWilliams type identities on the Lee and Euclidean weights for linear   codes over $\mathbb{Z}_{\ell}$
  Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}.$ Necessary and sufficient conditions for the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}$ are given. Some examples about such MacWilliams type identities are also presented. 
Yongsheng Tang, Shixin Zhu, Xiaoshan Kai
cs.IT math.IT