Dimensions of the hull of generalized Reed-Solomon codes

Let GRS$ _k(\boldsymbol{\alpha}, \boldsymbol{\upsilon}) $ be a $ k $-dimensional generalized Reed-Solomon (GRS) code over $ \mathbb{F}_q $ associated with $ \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_n) $ and $ \boldsymbol{\upsilon} = (\upsilon_1, \ldots, \upsilon_n) $. In this paper, we determ...

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Bibliographic Details
Published inAIMS mathematics Vol. 9; no. 6; pp. 13553 - 13569
Main Authors Huang, Jing, Liu, Jingge, Yu, Dong
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2024
Subjects
algebraic geometry code
generalized reed-solomon code
hull of a code
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ISSN2473-6988
2473-6988
DOI10.3934/math.2024661

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Summary:Let GRS$ _k(\boldsymbol{\alpha}, \boldsymbol{\upsilon}) $ be a $ k $-dimensional generalized Reed-Solomon (GRS) code over $ \mathbb{F}_q $ associated with $ \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_n) $ and $ \boldsymbol{\upsilon} = (\upsilon_1, \ldots, \upsilon_n) $. In this paper, we determined the dimension of the Euclidean hull GRS$ _k(\boldsymbol{\alpha}, \boldsymbol{\upsilon})\; \cap $ GRS$ _k(\boldsymbol{\alpha}, \boldsymbol{\upsilon})^\bot $, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2024661