Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908469397815296 |
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| author | Liu, Shengwei Liu, Hongwei Chen, Bocong |
| author_facet | Liu, Shengwei Liu, Hongwei Chen, Bocong |
| contents | Maximum distance separable (MDS) codes are considered optimal because the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are likely the generalized Reed-Solomon (GRS) codes. In 1989, Roth and Lempel constructed a type of MDS code that is not a GRS code (referred to as non-GRS). In 2017, Beelen et al. introduced twisted Reed-Solomon (TRS) codes and demonstrated that many MDS TRS codes are indeed non-GRS. Following this, the definition of TRS codes was generalized to the most comprehensive form, which we refer to as generalized twisted Reed-Solomon (GTRS) codes. In this paper, we prove that two families of GTRS codes are non-GRS and provide a systematic generator matrix for a class of GTRS codes. Inspired by the form of the systematic generator matrix for GTRS codes,we also present a construction of non-GRS MDS codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix Liu, Shengwei Liu, Hongwei Chen, Bocong Information Theory Maximum distance separable (MDS) codes are considered optimal because the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are likely the generalized Reed-Solomon (GRS) codes. In 1989, Roth and Lempel constructed a type of MDS code that is not a GRS code (referred to as non-GRS). In 2017, Beelen et al. introduced twisted Reed-Solomon (TRS) codes and demonstrated that many MDS TRS codes are indeed non-GRS. Following this, the definition of TRS codes was generalized to the most comprehensive form, which we refer to as generalized twisted Reed-Solomon (GTRS) codes. In this paper, we prove that two families of GTRS codes are non-GRS and provide a systematic generator matrix for a class of GTRS codes. Inspired by the form of the systematic generator matrix for GTRS codes,we also present a construction of non-GRS MDS codes. |
| title | Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix |
| topic | Information Theory |
| url | https://arxiv.org/abs/2507.20559 |