The problem studied in this work is to determine the higher weight spectra of the Projective Reed–Muller codes associated to the Veronese 3-fold V in PG(9, q), which is the image of the quadratic Veronese embedding of PG(3, q) in PG(9, q). We reduce the problem to the following combinatorial problem in finite geometry: For each subset S of V, determine the dimension of the linear subspace of PG(9, q) generated by S. We develop a systematic method to solve the latter problem. We implement the method for q = 3, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field Fqwill be treated in a future work.