Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes
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| Format: | Preprint |
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2025
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| _version_ | 1866908605840621568 |
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| author | Couvreur, Alain Pratihar, Rakhi |
| author_facet | Couvreur, Alain Pratihar, Rakhi |
| contents | In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $\mathbb{L} / \mathbb{K}$, these codes are defined as some specific $\mathbb{L}$-subspaces of the twisted group algebra $\mathbb{L} [\textrm{G}]$. We investigate the decoding of such codes in the "binary" case, \emph{i.e.,} when $\textrm{G} = (\mathbb{Z}/2\mathbb{Z})^m$. Our approach takes its inspiration from the decoding of Hamming metric binary Reed-Muller codes using their recursive Plotkin "$(u ~|~ u+v)$" structure. If our recursive algorithm restricts to a specific subclass of rank metric Reed-Muller codes, its asymptotic complexity beats that of the recently proposed decoding algorithm for arbitrary rank metric Reed-Muller codes based on Dickson matrices. Also, this decoder is of completely different nature and leads a natural rank metric counterpart of the Plotkin construction. To illustrate this, we also propose a generic Plotkin-like construction for matrix rank metric codes with an associate decoder, which can be applied to any pair of codes equipped with an efficient decoder. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes Couvreur, Alain Pratihar, Rakhi Information Theory In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $\mathbb{L} / \mathbb{K}$, these codes are defined as some specific $\mathbb{L}$-subspaces of the twisted group algebra $\mathbb{L} [\textrm{G}]$. We investigate the decoding of such codes in the "binary" case, \emph{i.e.,} when $\textrm{G} = (\mathbb{Z}/2\mathbb{Z})^m$. Our approach takes its inspiration from the decoding of Hamming metric binary Reed-Muller codes using their recursive Plotkin "$(u ~|~ u+v)$" structure. If our recursive algorithm restricts to a specific subclass of rank metric Reed-Muller codes, its asymptotic complexity beats that of the recently proposed decoding algorithm for arbitrary rank metric Reed-Muller codes based on Dickson matrices. Also, this decoder is of completely different nature and leads a natural rank metric counterpart of the Plotkin construction. To illustrate this, we also propose a generic Plotkin-like construction for matrix rank metric codes with an associate decoder, which can be applied to any pair of codes equipped with an efficient decoder. |
| title | Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2510.19095 |