循环码的重量分布不仅反映了码的纠错能力, 而且有助于计算发现和纠 错的概率, 因此循环码的重量分布一直是编码理论中的一个重要研究课题. 文章利用有限域上二次型理论, 选取域中特殊非平方元的技巧以及一些已知的结论, 确定了有限域~${\text{bF}}_p$ 上具有三个非零点${\alpha }^{-1}$, ${\alpha }^{-\frac{p^k+1}{2}}$, ${\alpha }^{-\frac{p^m-1}{2}}$ 的循环码的重量分布, 其中~$k$, $m$ 为正整数, $\alpha$ 为有限域~${\text{bF}}_{p^m}$ 的一个本原元.

The Weight distribution of a cyclic code not only gives the error correcting ability of the code, but also allows computation of the error probability of error detection and correction, and so it always has been a important research topic of coding theory. By using quadratic theory over finite fields, and the technique of selecting special non-squares of finite fields and some related known results, this paper has determined the weight distribution of a class of cyclic codes over a finite field ${\text{bF}}_p$ with three nonzeros~${\alpha }^{-1}$, ${\alpha }^{-\frac{p^k+1}{2}}$, ${\alpha }^{-\frac{p^m-1}{2}}$, where $k$ and $m$ are positive integers, and $\alpha$ is a primitive element in a finite field ${\text{bF}}_{p^m}$.