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

Abstract

In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: 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 $L$. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing $L$ to be an ${\mathbb{F}}_q$-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the $\Gamma \mathrm{L}$-class of $L$ and the number of inequivalent codes we can construct starting from it.