Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals (Communications in Mathematical Research, 2022)
Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd primeand $m$ is a positive integer. Recently, [17] and [35] determined the weight distributions of subfield codes with the form $$\mathcal{C}_f=\{(({\rm Tr}(af(x)+bx)+c)_{x\in \mathbb{F}_{p^m}},{\rm Tr}(a)):a,b\in \mathbb{F}_{p^m},c\in \mathbb{F}_p\}$$ for $f(x) = x^2$ and $f(x) = x^{p^k+1}$ , respectively, where $Tr(·)$ is the trace functionfrom $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$, and $k$ 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 [17, 35]. The weight distributionsof the constructed codes are determined by applying the theory of quadraticforms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Severalexamples show that some of our codes and their duals have the best knownparameters according to the code tables in [16]. The duals of some proposedcodes are optimal according to the Sphere Packing bound if $p≥ 5$.