Binary Optimal Codes from \({\mathbb {Z}}_{2}[u]{\mathbb {Z}}_{2}[u,v]\)-Additive Cyclic and Additive Constacyclic Codes

Let \({\mathfrak {R}}={\mathbb {Z}}_{2}+u{\mathbb {Z}}_{2}\), where \(u^2=0\), and \({\textbf {S}}={\mathbb {Z}}_{2}+u{\mathbb {Z}}_{2}+v{\mathbb {Z}}_{2}+uv{\mathbb {Z}}_{2}\), where \(u^{2}=v^{2}=0\), \(uv=vu\). In this article, we study \({\mathfrak {R}} {\textbf {S}}\)-additive cyclic, additive constacyclic, and additive dual codes. We find the structural properties of these codes. The code C is characterized as an \({\textbf {S}}[y]\)-submodules of the ring \({\textbf {S}}_{\beta _{1},\beta _{2}}={{\mathfrak {R}}[y]/\langle y^{\beta _{1}}-1\rangle }\times {{\textbf {S}}[y]/\langle y^{\beta _{2}}-1\rangle }\). We define the extended Gray map \(\Psi _{1}:{\mathfrak {R}}^{\beta _{1}}\times {\textbf {S}}^{\beta _2}\longrightarrow {\mathbb {Z}}_{2}^{n}\) and use this map to find the binary images with good parameters. We also obtain the minimal generating polynomials and minimal spanning sets of the above-mentioned codes. Further, we provide some examples to support of \({\mathfrak {R}} {\textbf {S}}\)-additive cyclic codes. Finally, we present a Table 1 of optimal binary codes.