Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function $\Gamma (k,q)$ , that is defined as the largest number of nonzero weights a cyclic code of dimension $k$ over $\mathbb {F}_{q}$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$ -ary, as well as in the $q$ -ary Hamming code are determined, two difficult results of independent interest.

中文翻译：

---

一个循环码可以有多少个权重？

给出了给定长度、维数和字母表的循环码中最大权重数的上限和下限。考虑不可约循环码的应用。为特殊循环码（这里称为强循环码）给出了更清晰的上限，其非零码字的周期等于码的长度。渐近线是在函数上推导出来的 $\Gamma (k,q)$ ，即定义为维数循环码的最大非零权重数 $千$ 超过 $\mathbb {F}_{q}$ 可以有，并勾勒出计算它的算法。Reed-Muller 代码的一些无限族中的非零权重，二进制或 $q$ -ary，以及在 $q$ -ary 汉明码确定，两个独立兴趣的困难结果。