Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes
Vito Napolitano, Olga Polverino, Paolo Santonastaso, Ferdinando Zullo · Advances in Mathematics of Communications · 2022
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.