On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes
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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.
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