| Recently, with a major breakthroughs in coding theory over finite field, the property of codes over finite rings inspire a great interest of code researchers. In this paper, we introduced a generalized Gray map of Fq +uFq +… +uk-1Fq over finite rings, studied the covering radius about homogeneous distance of its linear codes with length n Period distribution of cyclic codes and its dual codes over Fq +uFq +…+uk-1Fq are discussed. Therefore, we also studied the cyclic codes over finite Non-chain ring F2 + uF2 + vF2 + uvF2. The details are given as follows:(1) gives the definition of homogeneous distance over the ring Fq +UFq +u2Fq +…uk-1Fq, and fred the way in study the covering radius of the ring by the generalized Gray map that we have been constructed .several property were obtained, and some upper and lower bounds on the covering radius were described.(2) period of cyclic codes over Fq +uFq +…um-1Fq was defined. And the necessary and sufficient condition of period r was studied. Obtained the period distribution over this ring by its generate polynomial of cyclic codes. Furthermore, We obtained the relationship and accurate counting formula of period distribution between cyclic codes and its dual codes.(3) we introduce the generator polynomial of cyclic codes over finite non-chain ring F2 + uF2 + vF2 + uvF2 , defined a Lee distance of the ring. Obtained the related property of cyclic codes and its dual codes by the Gray map we defined. Finally, we get the minimum Hamming weight of its Gray image. |
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