完全非线性函数的线性码的子域码及其构造
Subfield codes of linear codes from perfect nonlinear functions and their duals
摘要：
设$\mathbb{F}_{p^m}$是具有$p^m$元素的有限域，其中$p$是奇素数，$m$是正整数.最近，\cite{Hengar}和\cite{Wang2020}分别为$f(x)=x^2$和$f(x)=x^{p^k+1}$确定了具有$$\mathcal{C}_f=\left\{\left(\left( {\rm Tr}_1^m(a f(x)+bx)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right)\, : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\}$$形式的子域码的权重分布，其中$k$是非负整数。本文进一步研究了$\mathbb{F}_{p^m}$上的完备非线性函数$f(x)$的子域码$\mathcal{C}_f$，推广了\cite{Hengar,Wang2020}中的一些结果.应用二次型理论和有限域上完全非线性函数的性质，确定了所构造码的重量分布。此外，还确定了这些码的编码参数。几个例子表明，我们的一些代码和他们的代码具有最好的已知参数相对于代码表在\cite{MGrassl}。当$p\geq 5$满足时，某些码的收敛性相对于球填充界是最优的.
英文摘要：
Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. Recently, \cite{Hengar} and \cite{Wang2020} determined the weight distributions of subfield codes with the form $$\mathcal{C}_f=\left\{\left(\left( {\rm Tr}_1^m(a f(x)+bx)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right)\, : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\}$$ for $f(x)=x^2$ and $f(x)=x^{p^k+1}$, respectively, where $k$ is a nonnegative integer. In this paper, we further investigate the subfield code $\mathcal{C}_f$ for $f(x)$ being a known perfect nonlinear function over $\mathbb{F}_{p^m}$ and generalize some results in \cite{Hengar,Wang2020}. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters with respect to the code tables in \cite{MGrassl}. The duals of some proposed codes are optimal with respect to the Sphere Packing bound if $p\geq 5$.
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。
