MacWilliams Identities of the Linear Codes Over \({\mathbb {F}}_q\times ({\mathbb {F}}_q+v{\mathbb {F}}_q)\)

Let R be the \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ -algebra \({\mathbb {F}}_q\times ({\mathbb {F}}_q+v{\mathbb {F}}_q)\) ${\mathbb{F}}_q\times ({\mathbb{F}}_q+v{\mathbb{F}}_q)$ of order \(q^3\) $q^3$ where \(v^2=v\) $v^2=v$ and \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ is a finite field of q elements. We study the MacWilliams identities of the linear codes over R related to complete, Hamming, symmetric, Gray and Lee weight enumerators.