中文摘要：

循环码是线性分组码中最重要的一个子类,由于其具有代数结构清晰、编译码简单且易于实现,被广泛地应用于通信系统和储存设备中。目前,大部分已有的研究工作最多只能实现三元最优循环码,对五元循环码的研究工作较少。对一类五元最优循环码C（（1,e,t））进行研究。首先,给出一种有效且快速判断五元循环码C（（1,e,t））是否最优的方法;其次,基于提出的方法得到当e=5-k＋1及e=5-m-2时,循环码C（（1,e,t））为最优循环码;最后,基于有限域F（5-m）中的完全非线性函数,构造一类具有参数[5-m-1,5-m-2m-2,4]的五元最优循环码。

英文摘要：

Cyclic codes are an extremely important subclass of linear codes.They are widely used in the communication systems and data storage systems because they have efficient encoding and decoding algorithm.Until now,how to construct the optimal ternary cyclic codes has received a lot of attention and much progress has been made.However,there is less research about the optimal quinary cyclic codes.Firstly,an efficient method to determine if cyclic codesC（（1,e,t）） were optimal codes was obtained.Secondly,based on the proposed method,when the equation e=5-k＋1 or e=5-m-2 hold,the theorem that the cyclic codesC（（1,e,t）） were optimal quinary cyclic codes was proved.In addition,perfect nonlinear monomials were used to construct optimal quinary cyclic codes with parameters[5-m-1,5-m-2m-2,4] optimal quinary cyclic codes over F（5-m）.