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

Abstract

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 $p\geq 5$.