Higher weight spectra and Betti numbers of Reed-Muller codes RM_q(2,2)

2024/08/05 by Ghorpade, Sudhir R., Johnsen, Trygve, Ludhani, Rati +1
#11B65 #94B27 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Primary: 94B05 #Secondary: 14G50

paper · doi:10.48550/arxiv.2408.02548

Abstract

We determine the higher weight spectra of q-ary Reed-Muller codes C_q=RM_q(2,2) for all prime powers q. This is equivalent to finding the usual weight distributions of all extension codes of C_q over every field extension of F_q of finite degree. To obtain our results we will utilize well-known connections between these weights and properties of the Stanley-Reisner rings of a series of matroids associated to each code C_q. In the process, we are able to explicitly determine all the graded Betti numbers of matroids associated to C_q and its elongations.

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