On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields

Consider a finite field Fq and positive integers d, m, r with 1⩽r⩽(m+d / d ). Let Sd(m) be the Fq vector space of all homogeneous polynomials of degree d in X0,⋯,Xm Let er(d,m) be the maximum number of Fq-rational points in the vanishing set of W as W varies through all subspaces of Sd(m) of dimension r. Beelen, Datta, and Ghorpade conjectured an exact formula of er(d,m) when q⩾d+1. We prove that their conjectured formula is true when q is sufficiently large in terms of m, d, r. The problem of determining er(d,m)$ is equivalent to the problem of computing the rth generalized Hamming weight of the projective Reed–Muller code PRMq(d,m). It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension r.