论文标题
二进制线性代码，重量很少。
Binary linear codes with few weights from two-to-one functions
论文作者
论文摘要
在本文中，我们在二进制线性代码的两个通用构造中应用了$ \ mathbb {f} _ {2^n} $的两对一功能。我们以两种形式考虑两到一的函数：（1）广义二次函数；和（2）$ \ left（x^{2^t}+x \ right）^e $ with $ \ gcd（t，n）= 1 $和$ \ gcd \ left（e，2^n-1 \ right）= 1 $。基于对这些函数或其相关主体的WALSH变换的研究，我们介绍了许多非零重量的线性代码，包括一个重量，三个重量，四个重量，四个重量和五个权重。确定了一个重量和三个重量的拟议代码的重量分布。此外，我们讨论了构造代码双重双重的最小距离，并表明其中一些达到了球形堆积。 {此外，几个示例表明我们的某些代码是最佳的，有些是最著名的参数。}
In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$ with $\gcd(t, n)=1$ and $\gcd\left(e, 2^n-1\right)=1$. Based on the study of the Walsh transforms of those functions or their related-ones, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. { Moreover, several examples show that some of our codes are optimal and some have the best known parameters.}
