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Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is anodd prime and $m$ is a positive integer. Recently, \cite{Hengar} and\cite{Wang2020} determined the weight distributions of subfield codes with theform  $$\mathcal{C}_f=\left\{\left(\left( {\rm Tr}_1^m(a f(x)+bx)+c\right)_{x \in\mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right)\, : \, a,b \in \mathbb{F}_{p^m}, c\in \mathbb{F}_p\right\}$$ 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 thesubfield code $\mathcal{C}_f$ for $f(x)$ being a known perfect nonlinearfunction over $\mathbb{F}_{p^m}$ and generalize some results in\cite{Hengar,Wang2020}.  The weight distributions of the constructed codes are determined by applyingthe theory of quadratic forms and the properties of perfect nonlinear functionsover finite fields. In addition, the parameters of the duals of these codes arealso determined. Several examples show that some of our codes and their dualshave the best known parameters with respect to the code tables in\cite{MGrassl}. The duals of some proposed codes are optimal with respect tothe Sphere Packing bound if $p\geq 5$.

Subfield codes of linear codes from perfect nonlinear functions and their duals