On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]\mathbb{Z}_{2}[u,v]$-additive cyclic codes and their application in obtaining optimal codes

Abstract

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{2}}{=}\mathbb{Z}_{2}+u\mathbb{Z}_{2}+v\mathbb{Z}_{2}$, where $u^{2}{=}v^{2}=0{=}uv{=}vu$. In this article, we study  $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{2}}$-additive cyclic, additive constacyclic and additive dual codes and find the structural properties of these codes. The additive cyclic codes are characterized as $\mathfrak{R_{2}}[y]$-submodules of a ring $\mathcal{S}_{\beta_{1},\beta_{2}, \beta_{3}}={\mathbb{Z}_{2}[y]/\langle y^{\beta_{1}}-1\rangle} \times{\mathfrak{R_{1}}[y]/\langle y^{\beta_{2}}-1\rangle}\times{\mathfrak{R_{2}}[y]/\langle y^{\beta_{3}}-1\rangle}$. The extended Gray map is represented by $\Psi_{1}:\mathbb{Z}_{2}^{\beta_{1}}\times\mathfrak{R_{1}}^{\beta_{2}}\times    \mathfrak{R_{2}}^{\beta_{3}}\longrightarrow\mathbb{Z}_{2}^{\beta_{1}+2\beta_{2}+4\beta_{3}}$ and is utilized to construct the binary codes with good parameters. The minimal generating polynomials and smallest spanning sets of the above specified codes are obtained. We  also establish  the relationship between the minimal polynomials of additive cyclic  codes and their duals.  Further, we provide some examples  that support  our main results. Finally, the  optimal binary codes are determined in Table\ref{Table1}.