Dimensions of the hull of generalized Reed-Solomon codes

Abstract

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.

Keywords

• hull of a code
• generalized reed-solomon code
• algebraic geometry code