  Following Johnsen and Verdure (2013), we can associate to any linear code $C$
an abstract simplicial complex and in turn, a Stanley-Reisner ring $R_C$. The
ring $R_C$ is a standard graded algebra over a field and its projective
dimension is precisely the dimension of $C$. Thus $R_C$ admits a graded minimal
free resolution and the resulting graded Betti numbers are known to determine
the generalized Hamming weights of $C$. The question of purity of the minimal
free resolution of $R_C$ was considered by Ghorpade and Singh (2020) when $C$
is the generalized Reed-Muller code. They showed that the resolution is pure in
some cases and it is not pure in many other cases. Here we give a complete
characterization of the purity of graded minimal free resolutions of
Stanley-Reisner rings associated to generalized Reed-Muller codes of an
arbitrary order.
