Betti Numbers and Higher Weight Spectra of Reed-Muller Codes $RM_q(2,2)$
For coding theorists and algebraic geometers, this provides a complete combinatorial description of these codes, though the result is incremental as it extends known methods to a specific case.
The paper determines all Betti numbers and higher weight spectra of q-ary second-order Reed-Muller codes of length q^2, recovering previous results on counting curves over finite fields with prescribed intersection points.
We determine all the Betti numbers of the $q$-ary second order Reed-Muller codes of length $q^2$, 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 $\mathbb{A}^2$ under the quadratic Veronese embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$. Indeed, finding the higher weight spectra, of the Reed-Muller code considered here, corresponds to determining the number of $\mathbb{F}_q$-rational points on all possible sections of this affine Veronesean by linear subvarieties of $\mathbb{P}^5$.