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arXiv · 2406.03803

Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m)

Yueying Lou·Qichun Wang
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Abstract

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.

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Yueying Lou, Qichun Wang. 2024-06-06. Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m). https://arxiv.org/abs/2406.03803

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