Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$
<|reference_start|>Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$: 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.<|reference_end|>
@article{ghorpade2024higher,
    title={Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$},
    author={Sudhir R. Ghorpade, Trygve Johnsen, Rati Ludhani, and Rakhi Pratihar},
    journal={arXiv preprint arXiv:2408.02548},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.02548},
    primaryClass={math.CO cs.IT math.IT}
}