$\mathbb{Z}_{2}\mathbb{Z}_{2}[u]\mathbb{Z}_{2}[u^k]$-additive cyclic codes and construction of optimal binary codes

Abstract

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{u^{k}}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}+\cdots+u^{k-1 }\mathbb{Z}_{2}$, where $u^{k}=0$. In this article, we study  $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{u^k}}$-additive cyclic codes and their structural properties. The additive cyclic codes are characterized as $\mathfrak{R_{u^k}}[y]$-submodules of the ring $$\mathcal{S}_{\beta_{1},\beta_{2}, \beta_{3}}={\mathbb{Z}_{2}[y]/\langle y^{\beta_{1}}-1\rangle} \times{\mathfrak{R_{1}}[y]/\langle y^{\beta_{2}}-1\rangle}\times{\mathfrak{R_{u^k}}[y]/\langle y^{\beta_{3}}-1\rangle}.$$ The extended Gray map is represented by $\Psi_{1}:\mathbb{Z}_{2}^{\beta_{1}}\times\mathfrak{R_{1}}^{\beta_{2}}\times    \mathfrak{R_{u^k}}^{\beta_{3}}\longrightarrow\mathbb{Z}_{2}^{\beta_{1}+2\beta_{2}+k\beta_{3}}$ 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  minimal generating polynomials of additive cyclic codes and their duals. Further, we construct optimal binary codes.