Pure Resolutions, Linear Codes, and Betti Numbers

02/05/2020

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by   Sudhir R. Ghorpade, et al.

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We consider the minimal free resolution of Stanley-Reisner rings associated to a linear code and give an intrinsic characterization of linear codes having a minimal free resolution. We use this characterization to quickly deduce the minimal free resolution of Stanley-Reisner ring associated to MDS codes as well as constant weight codes. Further, we compute the Betti numbers of the Stanley-Reisner ring associated to first order Reed-Muller codes and prove that resolutions of Stanley-Reisner ring associated to binary Reed-Muller codes are not pure in general. Using these results we determine the nature of the minimal free resolution of Stanley-Reisner rings corresponding to several known classes of two-weight codes