  We determine the higher weight spectra of $q$-ary Reed-Muller codes
$C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the
usual weight distributions of all extension codes of $C_q$ over every field
extension of $F_q$ of finite degree. To obtain our results we will utilize
well-known connections between these weights and properties of the
Stanley-Reisner rings of a series of matroids associated to each code $C_q$. In
the process, we are able to explicitly determine all the graded Betti numbers
of matroids associated to $C_q$ and its elongations.
