In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R1 = F2 ＋ u2 （u^2 = 1） to be a reversible cyclic code. W...In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R1 = F2 ＋ u2 （u^2 = 1） to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring R = F2 [u, v]/（u^2 -1, v^3 -v, uv- vu）, which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring /{ and 64 codons used in the amino acids of living organisms by introducing a Gray map from R to R1. Among others, binary images of cyclic codes over R are also investigated. As applications, some cyclic DNA codes over R using the Gray map are provided.展开更多
文摘In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R1 = F2 ＋ u2 （u^2 = 1） to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring R = F2 [u, v]/（u^2 -1, v^3 -v, uv- vu）, which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring /{ and 64 codons used in the amino acids of living organisms by introducing a Gray map from R to R1. Among others, binary images of cyclic codes over R are also investigated. As applications, some cyclic DNA codes over R using the Gray map are provided.
