On some quadratic APN functions

A construction of APN functions using the bent function B(x,y)=xy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(x,y)=xy$$\end{document} is proposed in Carlet (Des Codes Cryptogr 59:89–109, 2011). At this time, two families of APN functions using this construction are known, that is, the family of Carlet (2011) and the family of Zhou and Pott (Adv Math 234:43–60, 2013). In this note, we propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on F28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {F}}}_{2^8}$$\end{document}. We also propose a family of presemifields and determined the middle, left, right nuclei and the center of the associated semifields.