2021/01/30 by Sudhir R. Ghorpade, Ghorpade, Sudhir R., Rati Ludhani +1
Computer Science · Mathematics · #13D02 #94B05 #Algebraic structures and combinatorial models #Coding theory and cryptography #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.2102.00308
openalex publication_date 2021/01/30 · openalex created_date 2022/12/10 · openalex updated_date 2026/07/28
Following Johnsen and Verdure (2013), we can associate to any linear code C an abstract simplicial complex and in turn, a Stanley-Reisner ring RC. The ring RC is a standard graded algebra over a field and its projective dimension is precisely the dimension of C. Thus RC 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 RC 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.