Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

Abstract

In this paper we consider two pointsets in \begin{document}$ \mathrm{PG}(2,q^n) $\end{document} arising from a linear set \begin{document}$ L $\end{document} of rank \begin{document}$ n $\end{document} contained in a line of \begin{document}$ \mathrm{PG}(2,q^n) $\end{document}: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set \begin{document}$ L $\end{document}. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing \begin{document}$ L $\end{document} to be an \begin{document}$ {\mathbb F}_{q} $\end{document}-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the \begin{document}$ \Gamma\mathrm{L} $\end{document}-class of \begin{document}$ L $\end{document} and the number of inequivalent codes we can construct starting from it.