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arXiv · 2002.01799

Pure Resolutions, Linear Codes, and Betti Numbers

Sudhir R. Ghorpade·Prasant Singh
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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.

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Sudhir R. Ghorpade, Prasant Singh. 2020-02-05. Pure Resolutions, Linear Codes, and Betti Numbers. https://doi.org/10.1016/j.jpaa.2020.106385

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