Let C n−1 (n, q) be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG(n, q). Recently, Polverino and Zullo [12] proved that within this code, all non-zero code words of weight at most 2q n−1 are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when q > 17 and q / ∈ {25, 27, 29, 31, 32, 49, 121}, all code words of weight at most (4q − √ 8q − 33 2 )q n−2 are linear combinations of incidence vectors of hyperplanes through a fixed (n − 3)-space. Depending on the omitted value for q, we can lower the bound on the weight of c to obtain the same results.
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the
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