校验多项式含两个零的非投射循环码
Non-projective cyclic codes whose check polynomial contains two zeros
摘要：
令$n\geq 3$是正整数，并且令$\mathbb{F}_{q^k}$是$x^n-1$的分裂域。通过$γ$，我们表示$\mathbb{F}_{q^k}$的原始元素。令$C$是长度为$n$的循环码，其校验多项式包含两个零$γ^d$和$γ^{d+D}$，其中$de \mid (q-1)$、$e>1$和$D=(q^k-1)/e$。这个循环码族不是投射的。许多作者研究了这些码的重量分布的某些参数。在本文中，我们证明了这些码从来没有两个重量码。这一结果将加强Vega的一个猜想，即所有的二重循环码都是“已知的”。
英文摘要：
Let $n\geq 3$ be a positive integer and let $\mathbb{F}_{q^k}$ be the splitting field of $x^n-1$. By $γ$ we denote a primitive element of $\mathbb{F}_{q^k}$. Let $C$ be a cyclic code of length $n$ whose check polynomial contains two zeros $γ^d$ and $γ^{d+D}$, where $de \mid (q-1)$, $e>1$ and $D=(q^k-1)/e$. This family of cyclic codes is not projective. Many authors have studied the weight distribution of these codes for certain parameters. In this paper, we prove that these codes are never two-weight codes. This result would strengthen a conjecture by Vega which states that all two-weight cyclic codes are the "known" ones.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
