In this paper, a class of $p$ -ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over $\mathbb {F}_{p^{m}}$ and $(m,s)$ -bent functions from $\mathbb {F}_{2^{m}}$ to $\mathbb {F}_{2^{s}}$ , where $p$ is an odd prime and $m, s$ are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed $(m,s)$ -bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction.

中文翻译：

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有限域上完美非线性函数的线性代码

在本文中，一类 $p$ 凭借完美非线性函数的性质，分别提出了-ary 3权重线性码和一类二进制2权重线性码。 $\mathbb {F}_{p^{m}}$ 和 $(m,s)$ -bent 函数来自 $\mathbb {F}_{2^{m}}$ 至 $\mathbb {F}_{2^{s}}$ ， 在哪里 $p$ 是一个奇素数并且 $m, s$ 是正整数。权重分布完全由弱正则弯曲函数的沃尔什变换的符号和所用原像的大小决定 $(m,s)$ -bent 函数分别在零点。作为一个特例，我们的构造得到了一类满足 Griesmer 界的最优线性码。