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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2508.18845 |
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| _version_ | 1866914240741244928 |
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| author | Li, Yang Lu, Zhenliang Ling, San Zhu, Shixin Lam, Kwok Yan |
| author_facet | Li, Yang Lu, Zhenliang Ling, San Zhu, Shixin Lam, Kwok Yan |
| contents | Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs Li, Yang Lu, Zhenliang Ling, San Zhu, Shixin Lam, Kwok Yan Information Theory Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results. |
| title | Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs |
| topic | Information Theory |
| url | https://arxiv.org/abs/2508.18845 |