Constacyclic codes over $${{\mathbb {Z}}_2[u]}/{\langle u^2\rangle }\times {{\mathbb {Z}}_2[u]}/{\langle u^3\rangle }$$ and the MacWilliams identities

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.