Pré-Publication, Document De Travail Année : 2025

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

Résumé

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.

Dates et versions

hal-04909589 , version 1 (24-01-2025)

Identifiants

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Sudhir R. Ghorpade, Trygve Johnsen, Rati Ludhani, Rakhi Pratihar. Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$. 2025. ⟨hal-04909589⟩
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