Betti Numbers and Higher Weight Spectra of Reed–Muller Codes \({\textrm{RM}}_{{q}}{(2,2)}\)

Abstract. We determine all the Betti numbers of the [Formula: see text]-ary second order Reed–Muller codes of length [Formula: see text], and also of the elongations of matroids associated to these codes. We then use it to determine the higher weight spectra of these codes. As a special case, we recover some results of Kaplan and Matei about counting certain curves over finite fields with prescribed rational intersection points. In geometric terms, our results relate to the affine Veronesean by which we mean the image of the affine plane [Formula: see text] under the quadratic Veronese embedding of [Formula: see text] in [Formula: see text]. Indeed, finding the higher weight spectra of the Reed–Muller code considered here corresponds to determining the number of [Formula: see text]-rational points on all possible sections of this affine Veronesean by linear subvarieties of [Formula: see text].