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) $(x^{2^{t}}+x)^{e}$ with $\gcd (t, n)=\gcd \left ({e, 2^{n}-1}\right)=1$ . Based on the study of the Walsh transforms of those functions or their variants, 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, examples show that some codes in this paper have best-known parameters.

中文翻译：

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来自二对一函数的具有很少权重的二进制线性代码

在本文中，我们将二对一函数应用于 ${\mathbb F}_{2^{n}}$ 在二进制线性代码的两种通用结构中。我们以两种形式考虑二对一函数：（1）广义二次函数；和 (2) $(x^{2^{t}}+x)^{e}$ 和 $\gcd (t, n)=\gcd \left ({e, 2^{n}-1}\right)=1$ . 基于对这些函数或其变体的沃尔什变换的研究，我们提出了许多类具有很少非零权重的线性码，包括一权重、三权重、四权重和五权重。确定了具有一种权重和具有三种权重的建议代码的权重分布。此外，我们讨论了所构造代码的对偶的最小距离，并表明其中一些实现了球体填充边界。此外，示例表明本文中的一些代码具有最知名的参数。