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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.03803 |
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| _version_ | 1866910474778443776 |
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| author | Lou, Yueying Wang, Qichun |
| author_facet | Lou, Yueying Wang, Qichun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m) Lou, Yueying Wang, Qichun Information Theory 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. |
| title | Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m) |
| topic | Information Theory |
| url | https://arxiv.org/abs/2406.03803 |