MacWilliams type identities on the Lee and Euclidean weights for linear codes over Zℓ

Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over the ring Zℓ. Necessary and sufficient conditions for the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over Zℓ are given. Some examples about such MacWilliams type identities are also presented.