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 $3$-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 Fq will be treated in a future work.