In this article, we deal with additive codes over the Frobenius ring \({\mathcal {R}}_{2}{\mathcal {R}}_{3}:=\frac{{\mathbb {Z}}_{2}[u]}{\langle u^2 \rangle }\times \frac{{\mathbb {Z}}_{2}[u]}{\langle u^3 \rangle }\) . First, we study constacyclic codes over \({\mathcal {R}}_2\) and \({\mathcal {R}}_3\) and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over \({\mathcal {R}}_2{\mathcal {R}}_3\) . We use Gray maps to show that constacyclic codes over \({\mathcal {R}}_{2}{\mathcal {R}}_{3}\) are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these \({\mathcal {R}}_{2}{\mathcal {R}}_{3}\) -constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.
