Abstract
In this work, we focus on cyclic codes over the ring \({{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}\) , 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 \({({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)}\) and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \({{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}\) under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \({{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}\) in general.
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Communicated by T. Helleseth.
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Yildiz, B., Karadeniz, S. Cyclic codes over \({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}\) . Des. Codes Cryptogr. 58, 221–234 (2011). https://doi.org/10.1007/s10623-010-9399-3
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DOI: https://doi.org/10.1007/s10623-010-9399-3