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

Abstract

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 P G ( 9 , q ) , which is the image of the quadratic Veronese embedding of P G ( 3 , q ) in P G ( 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 P G ( 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 F q will be treated in a future work.