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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.21068 |
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| _version_ | 1866911535053406208 |
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| author | Xiong, Maosheng Yip, Chi Hoi |
| author_facet | Xiong, Maosheng Yip, Chi Hoi |
| contents | The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, $\text{BCH}(2,m)$, were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders $k$ has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of $ρ_2(\text{BCH}(2,m))$ and $ρ_3(\text{BCH}(2,m))$, and we establish a new lower bound for $ρ_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le ρ_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On generalized covering radii of binary primitive double-error-correcting BCH codes Xiong, Maosheng Yip, Chi Hoi Information Theory The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, $\text{BCH}(2,m)$, were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders $k$ has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of $ρ_2(\text{BCH}(2,m))$ and $ρ_3(\text{BCH}(2,m))$, and we establish a new lower bound for $ρ_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le ρ_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$. |
| title | On generalized covering radii of binary primitive double-error-correcting BCH codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2603.21068 |