Pure Resolutions, Linear Codes, and Betti Numbers
Abstract
We consider the minimal free resolutions of Stanley-Reisner rings associated to linear codes and give an intrinsic characterization of linear codes having a pure resolution. We use this characterization to quickly deduce the minimal free resolutions of Stanley-Reisner rings associated to MDS codes as well as constant weight codes. We also deduce that the minimal free resolutions of Stanley-Reisner rings of first order Reed-Muller codes are pure, and explicitly describe the Betti numbers. Further, we show that in the case of higher order Reed-Muller codes, the minimal free resolutions are almost always not pure. The nature of the minimal free resolution of Stanley-Reisner rings corresponding to several classes of two-weight codes, besides the first order Reed-Muller codes, is also determined.
Cite
@article{arxiv.2002.01799,
title = {Pure Resolutions, Linear Codes, and Betti Numbers},
author = {Sudhir R. Ghorpade and Prasant Singh},
journal= {arXiv preprint arXiv:2002.01799},
year = {2020}
}
Comments
Revised version; 25 pages; to appear in J. Pure Appl. Algebra