Cyclic codes over $${{mathbb{F}}_2+u{mathbb{F}}_2+v{mathbb{F}}_2+uv{mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring mathbbF₂+umathbbF₂+vmathbbF₂+uvmathbbF₂{{{mathbb{F}}_2+u{mathbb{F}}_2+v{mathbb{F}}_2+uv{mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (mathbbF₂+umathbbF₂+vmathbbF₂+uvmathbbF₂)/(xⁿ-1){({{mathbb{F}}_2+u{mathbb{F}}_2+v{mathbb{F}}_2+uv{mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over mathbbF₂+umathbbF₂+vmathbbF₂+uvmathbbF₂{{{mathbb{F}}_2+u{mathbb{F}}_2+v{mathbb{F}}_2+uv{mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over mathbbF₂+umathbbF₂+vmathbbF₂+uvmathbbF₂{{{mathbb{F}}_2+u{mathbb{F}}_2+v{mathbb{F}}_2+uv{mathbb{F}}_2}} in general.