Higher weight spectra of ternary codes associated to the quadratic Veronese 3-fold

The problem studied in this work is to determine the higher weight spectra of the Projective Reed–Muller codes associated to the Veronese [Formula: see text]-fold [Formula: see text] in [Formula: see text], which is the image of the quadratic Veronese embedding of [Formula: see text] in [Formula: see text]. We reduce the problem to the following combinatorial problem in finite geometry: For each subset [Formula: see text] of [Formula: see text], determine the dimension of the linear subspace of [Formula: see text] generated by [Formula: see text]. We develop a systematic method to solve the latter problem. We implement the method for [Formula: see text], and use it to obtain the higher weight spectra of the associated code. The case of a general finite field [Formula: see text] will be treated in a future work.