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

Abstract

<p style='text-indent:20px;'>In this paper we consider two pointsets in <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula> arising from a linear set <inline-formula><tex-math id="M3">\begin{document}$ L $\end{document}</tex-math></inline-formula> of rank <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> contained in a line of <inline-formula><tex-math id="M5">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula>: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set <inline-formula><tex-math id="M6">\begin{document}$ L $\end{document}</tex-math></inline-formula>. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing <inline-formula><tex-math id="M7">\begin{document}$ L $\end{document}</tex-math></inline-formula> to be an <inline-formula><tex-math id="M8">\begin{document}$ {\mathbb F}_{q} $\end{document}</tex-math></inline-formula>-linear set with a <i>short</i> weight distribution, then the associated codes have <i>few weights</i>. We conclude the paper by providing a connection between the <inline-formula><tex-math id="M9">\begin{document}$ \Gamma\mathrm{L} $\end{document}</tex-math></inline-formula>-class of <inline-formula><tex-math id="M10">\begin{document}$ L $\end{document}</tex-math></inline-formula> and the number of inequivalent codes we can construct starting from it.</p>

On the minimum size of linear sets

“…Recently, there has been an interest in linear sets admitting subspaces of complementary weights (see below for the definition), due to their application in coding theory, see e.g. [18,20,28]. Linear sets on the projective line admitting two points of complementary weights have been studied in [17] (see also [12,16]).…”

“…Linear sets are certain point sets in projective spaces, generalizing the notion of a subgeometry. They have proven themselves to be very useful in constructing interesting objects in projective spaces, such as blocking sets [26] and KM-arcs [9], and have been used to construct Hamming and rank metric codes [1,18,20,[23][24][25]27]. For a survey on linear sets, we refer the reader to [13,19].…”