Consider a finite field \(\mathbb {F}_q\) and positive integers d, m, r with \(1\leqslant r\leqslant \left( {\begin{array}{c}m+d\\ d\end{array}}\right) \) . Let \(S_d(m)\) be the \(\mathbb {F}_q\) vector space of all homogeneous polynomials of degree d in \(X_0,\dots ,X_m\) . Let \(e_r(d,m)\) be the maximum number of \(\mathbb {F}_q\) -rational points in the vanishing set of W as W varies through all subspaces of \(S_d(m)\) of dimension r. Beelen, Datta, and Ghorpade conjectured an exact formula of \(e_r(d,m)\) when \(q\geqslant d+1\) . We prove that their conjectured formula is true when q is sufficiently large in terms of m, d, r. The problem of determining \(e_r(d,m)\) is equivalent to the problem of computing the \(r^\textrm{th}\) generalized Hamming weight of the projective Reed–Muller code \(\textrm{PRM}_q(d,m)\) . 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.
