Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ positive integer. Recently, \cite{Hengar} \cite{Wang2020} determined the weight distributions of subfield codes form $$\mathcal{C}_f=\left\{\left(\left( {\rm Tr}_1^m(a f(x)+bx)+c\right)_{x \in \mathbb{F}_{p^m}}, Tr}_1^m(a)\right)\, : \, a,b \mathbb{F}_{p^m}, c \mathbb{F}_p\right\}$$ for $f(x)=x^2$ $f(x)=x^{p^k+1}$, respectively, $k$ nonnegative In this paper, we further investigate code $\mathcal{C}_f$ $f(x)$ being known perfect nonlinear function over generalize some results in \cite{Hengar,Wang2020}. The constructed are by applying theory quadratic forms properties functions fields. addition, parameters duals these also determined. Several examples show that our their have best respect to tables \cite{MGrassl}. The proposed optimal Sphere Packing bound if $p\geq 5$.