CONSTACYCLIC CODES OF LENGTH $2^s$ OVER $\mathbb F_{2}+u\mathbb F_{2}+v\mathbb F_{2}+uv\mathbb F_{2}$

In this paper, we investigate all constacyclic codes of length $2^s$ over $R=\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ , where $R$ is a local ring, but it is not a chain ring. First, by means of the Euclidean algorithm for polynomials over finite commutative local rings, we classify all cyclic and $(1+uv)$ -constacyclic codes of length $2^s$ over $R$ , and obtain their structure in each of those cyclic and $(1+uv)$ -constacyclic codes. Second, by using $(x-1)^{2^s}=u$ , we address the $(1+u)$ -constacyclic codes of length $2^s$ over $R$ , and get their classification and structure. Finally, by using similar discussion of $(1+u)$ -constacyclic codes, we obtain the classification and the structure of $(1+v), (1+u+uv), (1+v+uv), (1+u+v), (1+u+v+uv)$ -constacyclic codes of length $2^s$ over $R$ .