Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m)
<|reference_start|>Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m): The weight spectra of the Reed-Muller codes $RM(r,m)$ were unknown for $r=3,...,m-5$. In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of $RM(m-5,m)$ for $m\ge10$ using the Maiorana-McFarland construction, where the result was tried to be extended to $RM(m-6,m)$, but many problems occurred and much work needed to be done. In this paper, we propose a novel way of constructing Reed--Muller codewords and determine the weight spectrum of $RM(m-6,m)$ for $m\ge12$, which gives a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ for $c=6$. Moreover, we put forward a conjecture and verify it for some cases. If the conjecture is true, then that open question can be completely solved.<|reference_end|>
@article{lou2024determining,
    title={Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m)},
    author={Yueying Lou and Qichun Wang},
    journal={arXiv preprint arXiv:2406.03803},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.03803},
    primaryClass={cs.IT math.IT}
}