论文标题
来自完美非线性函数及其双重的线性代码的子场代码
Subfield codes of linear codes from perfect nonlinear functions and their duals
论文作者
论文摘要
令$ \ mathbb {f} _ {p^m} $是一个有限字段，带有$ p^m $元素，其中$ p $是一个奇怪的prime，$ m $是一个正整数。最近，\ cite {hengar}和\ cite {wang2020}确定了用表单的子场代码的重量分布 $ \ MATHCAL {C} _f = \ left \ {\ left（\ left（\ weled（{\ rm tr} _1^m（a f（a f（x）+bx）+c \ right）_ {x \ in \ in \ int \ mathbb {f} _ {f} _ {p^m}} \，a，b \ in \ mathbb {f} _ {p^m}，c \ in \ mathbb {f} _p} _p \ right \} $$对于$ f（x）= x^2 $ and $ f（x）= x^2 $和$ f（x）= x^{p^k+1} $，分别是$ k $ a $ k $ a n nonnone nonnenegate insege。在本文中，我们进一步研究了$ f（x）$的子场代码$ \ MATHCAL {C} _f $是$ \ Mathbb {f} _ {p^m} $上已知的完美非线性功能，并将某些结果推广到\ cite {Hengar，Wang2020202020}中。 构造代码的重量分布是通过应用二次形式的理论以及在有限场上的完美非线性函数的特性来确定的。另外，还确定了这些代码双重的参数。几个示例表明，我们的某些代码及其二元组具有相对于\ cite {mgrassl}中的代码表的最著名参数。如果$ p \ geq 5 $，则某些提出的代码的双重码相对于球形包装是最佳的。
Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. Recently, \cite{Hengar} and \cite{Wang2020} determined the weight distributions of subfield codes with the form $$\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 $p\geq 5$.
